122

Multiple Choice
Identify the choice that best completes the statement or answers the question.

____	1.	What is the value of  to the nearest thousandth?

A.	–1.150	C.	–1.524	
B.	1.150	D.	0.870	

____	2.	What is the value of  to the nearest thousandth?

A.	0.970	C.	–0.970	
B.	–0.242	D.	1.031	

____	3.	What is the value of  to the nearest thousandth?

A.	1.440	C.	1.390	
B.	–1.440	D.	0.694	

____	4.	What is the measure of the reference angle for an angle of  in standard position?

A.	298º	C.	28º	
B.	–62º	D.	62º	

____	5.	What is the measure of the reference angle for an angle of  in standard position?

A.	6º	C.	–186º	
B.	–6º	D.	84º	

____	6.	Which of these angles is coterminal with an angle of  in standard position?

A.	70º	C.	–70º	
B.	290º	D.	90º	

____	7.	Which of these angles is NOT coterminal with an angle of  in standard position?

A.	–530º	C.	550º	
B.	–170º	D.	370º	

____	8.	Which expression represents the measures of all the angles coterminal with  in standard position?

A.		C.		
B.		D.		

____	9.	Given , which statement is true for all possible values of q?

A.		
B.		
C.		
D.	 cannot be determined	

____	10.	What is the length of the arc that subtends a central angle of  in the unit circle?

A.	p units	C.	p units	
B.	p units	D.	p units	

____	11.	What is the length of the arc that subtends a central angle of  in the unit circle?

A.	p units	C.	p units	
B.	p units	D.	p units	

____	12.	What is the length of the arc that subtends a central angle of  in a circle with radius 4 units?

A.	p units	C.	p units	
B.	p units	D.	p units	

____	13.	What is the length of the arc that subtends a central angle of  in a circle with a radius of 3 units?

A.	p units	C.	p units	
B.	p units	D.	p units	

____	14.	In the unit circle, the length of the arc that subtends a positive central angle of q is p units. 
What is the measure of q in degrees?

A.	210°	C.	420°	
B.	210p°	D.	°	

____	15.	In a circle with radius 7 units, the length of the arc that subtends a positive central angle of q  is 
p units. What is the measure of q  in degrees?

A.	90°	C.	315°	
B.	45°	D.	45p°	

____	16.	What is  in radians?

A.	 radians	C.	 radians	
B.	60p radians	D.	 p radians	

____	17.	What is  in radians?

A.	p radians	C.	–120p radians	
B.	 radians	D.	 radians	

____	18.	What is  in radians?

A.	–490p radians	C.	p radians	
B.	 radians	D.	 radians	

____	19.	What is 2p radians in degrees?

A.	360°	C.	1°	
B.	115°	D.	6°	

____	20.	What is p radians in degrees? Give the answer to the nearest degree. 

A.	424°	C.	43°	
B.	135°	D.	2°	

____	21.	What is –2 radians in degrees? Give the answer to the nearest degree. 

A.	–360°	C.	–6°	
B.	–1°	D.	–115°	

____	22.	What is the value of  to the nearest hundredth?

A.	0.59	C.	1.70	
B.	–15.21	D.	–1.07	

____	23.	What is the value of  to the nearest hundredth?

A.	1.07	C.	0.93	
B.	–1.70	D.	47.75	

____	24.	In the unit circle, what is the length of the arc that subtends a central angle of  radians?
Give the answer to the nearest tenth.

A.	0.1 units	C.	1.0 units	
B.	60.0 units	D.	0.3 units	

____	25.	In a circle with radius 4 cm, what is the length of the arc that subtends a central angle of 
 radians? Give the answer to the nearest tenth.

A.	1.6 cm	C.	0.1 cm	
B.	0.4 cm	D.	0.5 cm	

____	26.	In a circle with radius 6 cm, what is the length of the arc that subtends a central angle of 3.5 radians? 
Give the answer to the nearest tenth.

A.	21.0 cm	C.	3.5 cm	
B.	0.6 cm	D.	66.0 cm	

____	27.	Which angle is NOT coterminal with an angle of p radians in standard position?

A.	p	C.	p	
B.	p	D.	p	

____	28.	Which function below has this graph?

A.		
B.		
C.		
D.	None of the above	

____	29.	Graph  using graphing technology.
What is the range of ?

A.		C.		
B.		D.		

____	30.	Graph  using graphing technology.
What is the y-intercept of the graph?

A.	0	C.		
B.		D.	1	

____	31.	Graph  using graphing technology.
Which of these equations is NOT the equation of an asymptote of ?

A.		C.		
B.		D.		

____	32.	Graph  using graphing technology.
Which of these values is NOT an x-intercept of ?

A.		C.		
B.		D.		

____	33.	Which function below has this graph?

A.		
B.		
C.		
D.	None of the above	

____	34.	Which characteristic differs among these three sinusoidal graphs?

A.	amplitude	B.	period	C.	phase shift	D.	centre line	

____	35.	Which set of functions describes these graphs?

A.	

	B.	

	C.	

	D.	

____	36.	What is the amplitude of the function ?

A.	–7	C.	14	
B.	7	D.	7 p	

____	37.	What is the amplitude of this sinusoidal function?

A.	3	C.	6	
B.	2p	D.	–3	

____	38.	What is the period of the function ?

A.		C.		
B.		D.		

____	39.	What is the period of this function?

A.		C.		
B.		D.		

____	40.	What is the phase shift of the function ?

A.		C.		
B.		D.		

____	41.	This graph is the image of  after a phase shift.
Which value below could represent the phase shift?

A.		C.		
B.		D.		

____	42.	Which function below describes this graph?

A.		C.		
B.		D.		

____	43.	Which function below describes this graph?

A.		C.		
B.		D.		

____	44.	Which function below describes this graph?

A.		C.		
B.		D.		

____	45.	Which function below describes this graph?

A.		C.		
B.		D.		

____	46.	Which number is NOT in the domain of ?

A.	p	C.	p	
B.	p	D.	p	

____	47.	Identify the transformations that would be applied to the graph of  to get the graph of .

A.	A vertical stretch by a factor of 7, and then a translation of  units left	
B.	A vertical stretch by a factor of , and then a translation of 7 units right	
C.	A vertical stretch by a factor of 7, and then a translation of  units right	
D.	A horizontal stretch by a factor of 7, and then a translation of  units right	

____	48.	Identify the transformations that would be applied to the graph of  to get the graph of .

A.	A vertical stretch by a factor of 3, and then a translation of 5 units down	
B.	A horizontal stretch by a factor of 5, and then a translation of  units right	
C.	A horizontal stretch by a factor of 3, and then a translation of 5 units down	
D.	A horizontal compression by a factor of , and then a translation of 5 units down	

____	49.	Identify the transformations that would be applied to the graph of  to get the graph of .

A.	A vertical stretch by a factor of 3, and then a translation of 5 units right	
B.	A vertical stretch by a factor of 5, and then a translation of 3 units down	
C.	A horizontal stretch by a factor of 5, and then a translation of 3 units down	
D.	A vertical stretch by a factor of 5, and then a translation of 3 units up	

____	50.	What is the amplitude of the graph of ?

A.	12	C.	10	
B.	8	D.	5	

____	51.	What is the equation of the centre line of the graph of the function ?

A.	2	C.		
B.	–5	D.	–35	

____	52.	What is the period of the function ?

A.		C.		
B.		D.		

____	53.	What is the range of the function ?

A.		C.		
B.		D.		

____	54.	Which equation below represents a cosine function with amplitude 1, period , equation of centre line , and phase shift ?

A.		C.		
B.		D.		

____	55.	Which graph below can be described by ?

A.		C.		
B.		D.		

____	56.	Which function below best describes this graph?

A.		C.		
B.		D.		

____	57.	Identify the transformations that would be applied to the graph of  to get the graph of .

A.	A horizontal compression by a factor of , and then a translation of  units right	
B.	A horizontal stretch by a factor of , and then a translation of  units up	
C.	A vertical compression by a factor of , and then a translation of  units up	
D.	A horizontal compression by a factor of , and then a translation of  units up	

____	58.	Identify the transformations that would be applied to the graph of  to get the graph of .

A.	A horizontal stretch by a factor of 3, and then a translation of  units right	
B.	A horizontal stretch by a factor of , and then a translation of 3 units left	
C.	A horizontal stretch by a factor of , and then a translation of 3 units right	
D.	A horizontal compression by a factor of , and then a translation of 3 units left	

____	59.	What is the period of the function ?

A.		C.		
B.		D.		

____	60.	What is the maximum value of the function ?

A.	2	C.	–5	
B.	–6	D.	–10	

____	61.	Which graph below can be described by ?

A.		C.		
B.		D.		

____	62.	Which function below best describes this graph?

A.		C.		
B.		D.		

____	63.	Suppose the function  models the height, y metres, of a seat on a Ferris wheel at any time x minutes after the wheel begins to rotate. How long does it take the Ferris wheel to complete one full rotation?

A.	11 min	C.	 min	
B.	15 min	D.	 min	

____	64.	Suppose the function  models the height, y metres, of a seat on a Ferris wheel at any time x minutes after the wheel begins to rotate. What is the diameter of the wheel?

A.	16 m	C.	7 m	
B.	35 m	D.	32 m	

____	65.	Suppose the function  models the height, y metres, of a seat on a Ferris wheel at any time x minutes after the wheel begins to rotate. After how many minutes is the seat at the top of the wheel for the first time?

A.	47 min	C.	3.5 min	
B.	14 min	D.	7 min	

Short Answer

	1.	Sketch the angle –380° in standard position, then identify the reference angle.

	2.	The angle 110° is shown below in standard position.
Sketch a coterminal angle between –360° and 0° on the same axes. 
Label the angle with its measure.

	3.	The angle 140° is shown below in standard position.
Sketch a coterminal angle between 360° and 720° on the same axes. 
Label the angle with its measure.

	4.	Determine the measures of all angles between  and  that are coterminal with an angle of  in standard position.

	5.	Determine the measures of all angles between  and  that are coterminal with an angle of  in standard position.

	6.	Determine the exact value of .

	7.	Determine the exact value of .

	8.	Determine the exact value of .

	9.	For the point P on the terminal arm of an angle q in standard position, determine the exact value of .

	10.	Given  , , determine the exact value of .

	11.	Given  , , determine the exact value of .

	12.	As a fraction of p, determine the length of the arc that subtends the central angle shown in this unit circle. 

	13.	As a fraction of p, determine the length of the arc that subtends the central angle shown in this circle. 

	14.	An arc with length p units subtends a central angle of 155° in a circle. 
Determine the radius of the circle.

	15.	An arc with length p units subtends a central angle of –355° in a circle.  
Determine the radius of the circle.

	16.	Sketch the angle p radians in standard position.

	17.	Sketch the angle –11.1 radians in standard position.

	18.	Label a sketch on the circle below to illustrate the relationship between the radius, r, of this circle and an angle of 1 radian.

	19.	Label a sketch on the circle below to illustrate the relationship between the radius, r, of this circle and the length of an arc that subtends an angle of q  radians.

	20.	Write an expression for the measures of all possible angles that are coterminal with an angle of .

	21.	Determine the measures of all angles that are coterminal with an angle of p in the domain .

	22.	Determine the exact value of .

	23.	An arc with length 7 cm is marked on the circumference of a circle with radius 5.5 cm. 
To the nearest tenth of a radian, determine the measure of the central angle subtended by the arc.

	24.	An arc with length 46 cm is marked on the circumference of a circle with radius 8 cm. Determine the area of the sector of the circle formed by the arc and the radii that intersect the endpoints of the arc. 

	25.	An arc with length 14 cm is marked on the circumference of a circle. The central angle subtended by the arc measures  radians. To the nearest tenth of a centimetre, determine the radius of the circle.

	26.	An arc is marked on the circumference of a circle. The central angle subtended by the arc measures 
 radians. The sector of the circle formed by the arc and the radii that intersect the endpoints of the arc has area 8.6 cm2. To the nearest tenth of a centimetre, determine the radius of the circle.

	27.	A wheel is rotating at 25 revolutions per minute. 
What is the exact angular velocity of the wheel in radians per second?

	28.	The graph of  is shown below for p.
Extend the graph for  and for .

	29.	Graph  using graphing technology.
How many zeros does the function  have over the domain pp?

	30.	Graph  using graphing technology.
Identify the zeros of the function  over the domain pp.

	31.	Graph  using graphing technology.
Determine the equations of the asymptotes of  over the domain pp.

	32.	The graph of  is shown below.
On the same grid, sketch the graph of .

	33.	The graph of  is shown below.
On the same grid, sketch the graph of .

	34.	The graph of  is shown below.
On the same grid, sketch the graph of .

	35.	The graph of  is shown below.
On the same grid, sketch the graph of .

	36.	What is the range of the function ?

	37.	Does this graph represent a periodic function? 

	38.	Write a general equation for the asymptotes of the graph of .

	39.	Identify the following characteristics of the graph of the sine function shown below.
•	amplitude
•	period
•	equation of the centre line
•	phase shift

	40.	Identify the following characteristics of the graph of  shown below.
•	amplitude
•	period
•	equation of the centre line	•	phase shift
•	zeros
•	domain	•	minimum value
•	maximum value
•	range	

	41.	Identify the following characteristics of the graph of the function +1.
•	amplitude
•	period
•	equation of the centre line
•	phase shift

	42.	Write an equation for a sine function with amplitude 2, period , equation of centre line , and phase shift .

	43.	Write an equation for a sine function with maximum value 8, minimum value 4, period , and phase shift .

	44.	Determine a possible equation for this graph. 

	45.	Determine a possible equation for this graph. 

	46.	Determine a possible equation for this graph. 

	47.	For the cosine function graphed below, determine the: 
•	amplitude
•	period
•	equation of the centre line
•	phase shift	•	maximum value
•	minimum value
•	domain
•	range 	

	48.	Write an equation that represents the sine function graphed below. 

	49.	Write an equation for a sinusoidal function that models the data graphed below.

	50.	A table fan has a mark on the tip of one blade. 
The equation  represents the height of the mark, y centimetres, above the table x seconds after the fan is turned on. What is the distance from the mark to the tip of the opposite blade?	

	51.	A table fan has a mark on the tip of one blade. 
The equation  represents the height of the mark, y centimetres, above the table x seconds after the fan is turned on. What is the height of the mark above the table when it is closest to the table?	

	52.	A bicycle with a stone stuck in the tread of one of its tires is pushed along the street. The function graphed below models the height, in inches, of the stone above the ground after x seconds. What is the height of the wheel axle above the ground? 

	53.	A bicycle with a stone stuck in the tread of one of its tires is pushed along the street. The function graphed below models the height, in inches, of the stone above the ground after x seconds. What is the diameter of the bicycle tire?

	54.	A bicycle with a stone stuck in the tread of one of its tires is pushed along the street. The function graphed below models the height, in inches, of the stone above the ground after x seconds. How often does the stone touch the ground?

	55.	These data show the fraction of the moon that was visible at midnight, Mountain Standard Time, in October, 2012. Identify the days when the maximum and minimum fractions of the moon were seen.

Problem

	1.	P is a terminal point of angle q  in standard position. 
Determine the exact values of the six trigonometric ratios for q.

	2.	Determine the exact values of the six trigonometric ratios for .

	3.	P is a terminal point of angle q in standard position. 
Determine all possible measures of q in the domain . 
Give the answers to the nearest degree.

	4.	Given , determine all possible measures of angle q in the domain .

	5.	Given , determine all possible measures of angle q  in the domain .
Give the answers to the nearest degree.

	6.	A Ferris wheel has diameter 38 m. Through what distance does a chair on the wheel travel when the wheel rotates 80°? Express the exact distance in terms of p.

	7.	A unicycle wheel has diameter 20 in. Suppose a positive angle of rotation corresponds to the wheel moving forward. Determine the distance and the direction the wheel will roll when it turns through an angle of –350°. Express the exact distance, in inches, in terms of p.

	8.	A fishing reel has diameter 4 cm. As the reel rotates, it pulls in a fishing line. Exactly how much fishing line is pulled in when the reel turns through an angle of 2000°? Express the exact length, in centimetres, in terms of p.

	9.	P is a terminal point of angle q  in standard position. 
Determine all possible measures of q  in the domain . 
Give the answers to the nearest tenth of a radian.

	10.	Given , determine all possible measures of angle q  in the domain .

	11.	Given , determine all possible measures of angle q  in the domain .
Give the answers to the nearest tenth of a radian.

	12.	A child skates on a circular rink with diameter 34 m. As she skates, she keeps one hand on the boards around the perimeter of the rink. It takes her 356 s to skate around the whole rink. 
a)	Assume the child skates at a constant speed. What is the length of the arc she skates in 59 s? Give the answer to the nearest tenth of a metre.
b)	After 59 s, the child skates in a straight line back to her starting point. How far does she skate across the ice? Give the answer to the nearest tenth of a metre.

	13.	In this diagram, both circles have centre O. 
The smaller circle has radius OD  m, and the larger circle has radius OB  m. Given , determine the exact area of the shaded region. 

	14.	Use a table of values to graph  for –2pp.

	15.	Use a table of values to graph  for –pp.

	16.	The graph of  is shown below.
Graph  on the same grid and explain how the graphs are related.

	17.	Graph .
Identify the amplitude, period, general expression for the zeros, general equation for the asymptotes, domain of the function, and range of the function.		

	18.	Graph . 
Identify the amplitude, period, general expression for the zeros, domain of the function, and range of the function.

	19.	Graph . 
Identify the amplitude, period, general expression for the zeros, general equation for the asymptotes, domain of the function, and range of the function.		

	20.	Graph . 
Identify the amplitude, period, general expression for the zeros, general equation for the asymptotes, domain of the function, and range of the function.		

	21.	Describe each transformation below.
a)	The graph of  is a transformation image of the graph of . 
b)	The graph of  is a transformation image of the graph of .

	22.	Write an equation in the form  for the graph below. Explain your reasoning.

	23.	The graph of  is shown below. On the same grid, sketch the graph of . 
Describe these characteristics of this function: amplitude, period, phase shift, equation of the centre line, domain, and range

	24.	Sketch the graph of .
Describe these characteristics of the function: amplitude, period, phase shift, equation of the centre line, domain, and range

	25.	Sketch the graph of .
Describe these characteristics of the function: amplitude, period, phase shift, equation of the centre line, domain, and range

	26.	Sketch the graph of a cosine function with amplitude 4, period , equation of centre line , and phase shift . Label the graph with its equation.

	27.	Sketch the graph of .
Describe these characteristics of the function: amplitude, period, phase shift, equation of the centre line, domain, and range

	28.	These data show the fraction of the moon that was visible at midnight, Mountain Standard Time, on alternate days in June, 2012. Graph the data, then write an equation of a sinusoidal function that models these data.

Day	Visible Fraction		
1	0.86		
3	0.98		
5	0.99		
7	0.89		
9	0.72		
11	0.52		
13	0.32		
15	0.16		
17	0.05		
19	0.00		
21	0.03		
23	0.13		
25	0.30		
27	0.52		
29	0.74		

	29.	A mark on the tip of one blade of a table fan is 24 cm from the 
axle, and the axle is 30 cm above the table. The fan is turned on when the mark has the position shown. The fan completes one clockwise rotation in  s. Write a sinusoidal function that represents the height of the mark, y centimetres, above the table x seconds after the fan is turned on.	

	30.	A car tire rolls onto a nail, which punctures the tire. The function graphed below models the height in centimetres of the head of the nail above the ground at any time t seconds after the nail punctured the tire. What is the average speed of the car to the nearest kilometre per hour? 

122
Answer Section

MULTIPLE CHOICE

	1.	ANS:	B	PTS:	1	DIF:	Easy	
REF:	6.1 Trigonometric Ratios for Any Angle in Standard Position	
LOC:	12.T3	TOP:	Trigonometry			KEY:	Procedural Knowledge

	2.	ANS:	A	PTS:	1	DIF:	Easy	
REF:	6.1 Trigonometric Ratios for Any Angle in Standard Position	
LOC:	12.T3	TOP:	Trigonometry			KEY:	Procedural Knowledge

	3.	ANS:	A	PTS:	1	DIF:	Easy	
REF:	6.1 Trigonometric Ratios for Any Angle in Standard Position	
LOC:	12.T3	TOP:	Trigonometry			KEY:	Procedural Knowledge

	4.	ANS:	D	PTS:	1	DIF:	Easy	
REF:	6.1 Trigonometric Ratios for Any Angle in Standard Position	
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

	5.	ANS:	A	PTS:	1	DIF:	Easy	
REF:	6.1 Trigonometric Ratios for Any Angle in Standard Position	
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

	6.	ANS:	A	PTS:	1	DIF:	Easy	
REF:	6.1 Trigonometric Ratios for Any Angle in Standard Position	
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

	7.	ANS:	D	PTS:	1	DIF:	Easy	
REF:	6.1 Trigonometric Ratios for Any Angle in Standard Position	
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

	8.	ANS:	D	PTS:	1	DIF:	Easy	
REF:	6.1 Trigonometric Ratios for Any Angle in Standard Position	
LOC:	12.T1	TOP:	Trigonometry			KEY:	Conceptual Understanding

	9.	ANS:	A	PTS:	1	DIF:	Easy	
REF:	6.1 Trigonometric Ratios for Any Angle in Standard Position	
LOC:	12.T3	TOP:	Trigonometry			KEY:	Conceptual Understanding

	10.	ANS:	D	PTS:	1	DIF:	Easy	
REF:	6.2 Angles in Standard Position and Arc Length	LOC:	12.T1
TOP:	Trigonometry			KEY:	Procedural Knowledge | Conceptual Understanding

	11.	ANS:	B	PTS:	1	DIF:	Easy	
REF:	6.2 Angles in Standard Position and Arc Length	LOC:	12.T1
TOP:	Trigonometry			KEY:	Procedural Knowledge | Conceptual Understanding

	12.	ANS:	D	PTS:	1	DIF:	Easy	
REF:	6.2 Angles in Standard Position and Arc Length	LOC:	12.T1
TOP:	Trigonometry			KEY:	Procedural Knowledge | Conceptual Understanding

	13.	ANS:	A	PTS:	1	DIF:	Easy	
REF:	6.2 Angles in Standard Position and Arc Length	LOC:	12.T1
TOP:	Trigonometry			KEY:	Procedural Knowledge | Conceptual Understanding

	14.	ANS:	A	PTS:	1	DIF:	Moderate	
REF:	6.2 Angles in Standard Position and Arc Length	LOC:	12.T1
TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding | Problem-Solving Skills

	15.	ANS:	B	PTS:	1	DIF:	Moderate	
REF:	6.2 Angles in Standard Position and Arc Length	LOC:	12.T1
TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding | Problem-Solving Skills

	16.	ANS:	D	PTS:	1	DIF:	Easy	REF:	6.3 Radian Measure
LOC:	12.T1	TOP:	Trigonometry			KEY:	Procedural Knowledge

	17.	ANS:	A	PTS:	1	DIF:	Easy	REF:	6.3 Radian Measure
LOC:	12.T1	TOP:	Trigonometry			KEY:	Procedural Knowledge

	18.	ANS:	C	PTS:	1	DIF:	Easy	REF:	6.3 Radian Measure
LOC:	12.T1	TOP:	Trigonometry			KEY:	Procedural Knowledge

	19.	ANS:	A	PTS:	1	DIF:	Easy	REF:	6.3 Radian Measure
LOC:	12.T1	TOP:	Trigonometry			KEY:	Procedural Knowledge

	20.	ANS:	B	PTS:	1	DIF:	Easy	REF:	6.3 Radian Measure
LOC:	12.T1	TOP:	Trigonometry			KEY:	Procedural Knowledge

	21.	ANS:	D	PTS:	1	DIF:	Easy	REF:	6.3 Radian Measure
LOC:	12.T1	TOP:	Trigonometry			KEY:	Procedural Knowledge

	22.	ANS:	C	PTS:	1	DIF:	Easy	REF:	6.3 Radian Measure
LOC:	12.T3	TOP:	Trigonometry			KEY:	Procedural Knowledge

	23.	ANS:	A	PTS:	1	DIF:	Easy	REF:	6.3 Radian Measure
LOC:	12.T3	TOP:	Trigonometry			KEY:	Procedural Knowledge

	24.	ANS:	C	PTS:	1	DIF:	Easy	REF:	6.3 Radian Measure
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding

	25.	ANS:	A	PTS:	1	DIF:	Easy	REF:	6.3 Radian Measure
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding

	26.	ANS:	A	PTS:	1	DIF:	Easy	REF:	6.3 Radian Measure
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding

	27.	ANS:	C	PTS:	1	DIF:	Moderate	REF:	6.3 Radian Measure
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

	28.	ANS:	C	PTS:	1	DIF:	Easy	
REF:	6.4 Graphing Trigonometric Functions		LOC:	12.T4
TOP:	Trigonometry			KEY:	Conceptual Understanding

	29.	ANS:	A	PTS:	1	DIF:	Easy	
REF:	6.4 Graphing Trigonometric Functions		LOC:	12.T4
TOP:	Trigonometry			KEY:	Conceptual Understanding

	30.	ANS:	D	PTS:	1	DIF:	Easy	
REF:	6.4 Graphing Trigonometric Functions		LOC:	12.T4
TOP:	Trigonometry			KEY:	Conceptual Understanding | Procedural Knowledge

	31.	ANS:	D	PTS:	1	DIF:	Moderate	
REF:	6.4 Graphing Trigonometric Functions		LOC:	12.T4
TOP:	Trigonometry			KEY:	Conceptual Understanding | Procedural Knowledge

	32.	ANS:	C	PTS:	1	DIF:	Moderate	
REF:	6.4 Graphing Trigonometric Functions		LOC:	12.T4
TOP:	Trigonometry			KEY:	Conceptual Understanding | Procedural Knowledge

	33.	ANS:	B	PTS:	1	DIF:	Difficult	
REF:	6.4 Graphing Trigonometric Functions		LOC:	12.T4
TOP:	Trigonometry			KEY:	Conceptual Understanding | Problem-Solving Skills

	34.	ANS:	D	PTS:	1	DIF:	Easy	REF:	6.5 Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			KEY:	Conceptual Understanding

	35.	ANS:	B	PTS:	1	DIF:	Easy	REF:	6.5 Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			KEY:	Conceptual Understanding

	36.	ANS:	B	PTS:	1	DIF:	Easy	REF:	6.5 Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding

	37.	ANS:	A	PTS:	1	DIF:	Easy	REF:	6.5 Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding

	38.	ANS:	D	PTS:	1	DIF:	Easy	REF:	6.5 Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			KEY:	Procedural Knowledge

	39.	ANS:	C	PTS:	1	DIF:	Moderate	REF:	6.5 Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			KEY:	Procedural Knowledge

	40.	ANS:	A	PTS:	1	DIF:	Easy	REF:	6.5 Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			KEY:	Procedural Knowledge

	41.	ANS:	A	PTS:	1	DIF:	Moderate	REF:	6.5 Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

	42.	ANS:	D	PTS:	1	DIF:	Easy	REF:	6.5 Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			KEY:	Conceptual Understanding

	43.	ANS:	B	PTS:	1	DIF:	Moderate	REF:	6.5 Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			KEY:	Conceptual Understanding

	44.	ANS:	C	PTS:	1	DIF:	Moderate	REF:	6.5 Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			KEY:	Conceptual Understanding

	45.	ANS:	C	PTS:	1	DIF:	Easy	REF:	6.5 Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			KEY:	Conceptual Understanding

	46.	ANS:	C	PTS:	1	DIF:	Moderate	REF:	6.5 Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

	47.	ANS:	C	PTS:	1	DIF:	Easy	
REF:	6.6 Combining Transformations of Sinusoidal Functions	LOC:	12.T4
TOP:	Trigonometry			KEY:	Procedural Knowledge

	48.	ANS:	C	PTS:	1	DIF:	Easy	
REF:	6.6 Combining Transformations of Sinusoidal Functions	LOC:	12.T4
TOP:	Trigonometry			KEY:	Procedural Knowledge

	49.	ANS:	B	PTS:	1	DIF:	Easy	
REF:	6.6 Combining Transformations of Sinusoidal Functions	LOC:	12.T4
TOP:	Trigonometry			KEY:	Procedural Knowledge

	50.	ANS:	D	PTS:	1	DIF:	Easy	
REF:	6.6 Combining Transformations of Sinusoidal Functions	LOC:	12.T4
TOP:	Trigonometry			KEY:	Procedural Knowledge

	51.	ANS:	B	PTS:	1	DIF:	Easy	
REF:	6.6 Combining Transformations of Sinusoidal Functions	LOC:	12.T4
TOP:	Trigonometry			KEY:	Procedural Knowledge

	52.	ANS:	A	PTS:	1	DIF:	Moderate	
REF:	6.6 Combining Transformations of Sinusoidal Functions	LOC:	12.T4
TOP:	Trigonometry			KEY:	Procedural Knowledge

	53.	ANS:	C	PTS:	1	DIF:	Moderate	
REF:	6.6 Combining Transformations of Sinusoidal Functions	LOC:	12.T4
TOP:	Trigonometry			KEY:	Procedural Knowledge | Conceptual Understanding

	54.	ANS:	A	PTS:	1	DIF:	Moderate	
REF:	6.6 Combining Transformations of Sinusoidal Functions	LOC:	12.T4
TOP:	Trigonometry			KEY:	Procedural Knowledge | Conceptual Understanding

	55.	ANS:	C	PTS:	1	DIF:	Moderate	
REF:	6.6 Combining Transformations of Sinusoidal Functions	LOC:	12.T4
TOP:	Trigonometry			KEY:	Procedural Knowledge | Conceptual Understanding

	56.	ANS:	B	PTS:	1	DIF:	Moderate	
REF:	6.6 Combining Transformations of Sinusoidal Functions	LOC:	12.T4
TOP:	Trigonometry			KEY:	Procedural Knowledge | Conceptual Understanding

	57.	ANS:	D	PTS:	1	DIF:	Easy	
REF:	6.7 Applications of Sinusoidal Functions		LOC:	12.T4
TOP:	Trigonometry			KEY:	Procedural Knowledge

	58.	ANS:	B	PTS:	1	DIF:	Easy	
REF:	6.7 Applications of Sinusoidal Functions		LOC:	12.T4
TOP:	Trigonometry			KEY:	Procedural Knowledge

	59.	ANS:	C	PTS:	1	DIF:	Moderate	
REF:	6.7 Applications of Sinusoidal Functions		LOC:	12.T4
TOP:	Trigonometry			KEY:	Procedural Knowledge

	60.	ANS:	B	PTS:	1	DIF:	Moderate	
REF:	6.7 Applications of Sinusoidal Functions		LOC:	12.T4
TOP:	Trigonometry			KEY:	Procedural Knowledge | Conceptual Understanding

	61.	ANS:	C	PTS:	1	DIF:	Moderate	
REF:	6.7 Applications of Sinusoidal Functions		LOC:	12.T4
TOP:	Trigonometry			KEY:	Procedural Knowledge | Conceptual Understanding

	62.	ANS:	C	PTS:	1	DIF:	Moderate	
REF:	6.7 Applications of Sinusoidal Functions		LOC:	12.T4
TOP:	Trigonometry			KEY:	Procedural Knowledge | Conceptual Understanding

	63.	ANS:	C	PTS:	1	DIF:	Moderate	
REF:	6.7 Applications of Sinusoidal Functions		LOC:	12.T4
TOP:	Trigonometry			KEY:	Procedural Knowledge | Conceptual Understanding

	64.	ANS:	D	PTS:	1	DIF:	Moderate	
REF:	6.7 Applications of Sinusoidal Functions		LOC:	12.T4
TOP:	Trigonometry			KEY:	Procedural Knowledge | Conceptual Understanding

	65.	ANS:	C	PTS:	1	DIF:	Moderate	
REF:	6.7 Applications of Sinusoidal Functions		LOC:	12.T4
TOP:	Trigonometry			KEY:	Procedural Knowledge | Conceptual Understanding

SHORT ANSWER

	1.	ANS:	

Reference angle: 20°

PTS:	1	DIF:	Easy	
REF:	6.1 Trigonometric Ratios for Any Angle in Standard Position	
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding | Communication

	2.	ANS:	

PTS:	1	DIF:	Easy	
REF:	6.1 Trigonometric Ratios for Any Angle in Standard Position	
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding | Communication

	3.	ANS:	

PTS:	1	DIF:	Easy	
REF:	6.1 Trigonometric Ratios for Any Angle in Standard Position	
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding | Communication

	4.	ANS:	
, , and 

PTS:	1	DIF:	Easy	
REF:	6.1 Trigonometric Ratios for Any Angle in Standard Position	
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

	5.	ANS:	
, , and 

PTS:	1	DIF:	Easy	
REF:	6.1 Trigonometric Ratios for Any Angle in Standard Position	
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

	6.	ANS:	

PTS:	1	DIF:	Moderate	
REF:	6.1 Trigonometric Ratios for Any Angle in Standard Position	
LOC:	12.T3	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding

	7.	ANS:	

PTS:	1	DIF:	Moderate	
REF:	6.1 Trigonometric Ratios for Any Angle in Standard Position	
LOC:	12.T3	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding

	8.	ANS:	

PTS:	1	DIF:	Moderate	
REF:	6.1 Trigonometric Ratios for Any Angle in Standard Position	
LOC:	12.T3	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding

	9.	ANS:	

PTS:	1	DIF:	Moderate	
REF:	6.1 Trigonometric Ratios for Any Angle in Standard Position	
LOC:	12.T3	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding

	10.	ANS:	

PTS:	1	DIF:	Moderate	
REF:	6.1 Trigonometric Ratios for Any Angle in Standard Position	
LOC:	12.T3	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding

	11.	ANS:	

PTS:	1	DIF:	Moderate	
REF:	6.1 Trigonometric Ratios for Any Angle in Standard Position	
LOC:	12.T3	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding

	12.	ANS:	
p

PTS:	1	DIF:	Easy	REF:	6.2 Angles in Standard Position and Arc Length
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding

	13.	ANS:	
p

PTS:	1	DIF:	Easy	REF:	6.2 Angles in Standard Position and Arc Length
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding

	14.	ANS:	
5 units

PTS:	1	DIF:	Moderate	REF:	6.2 Angles in Standard Position and Arc Length
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge | Problem-Solving Skills

	15.	ANS:	
3 units

PTS:	1	DIF:	Moderate	REF:	6.2 Angles in Standard Position and Arc Length
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge | Problem-Solving Skills

	16.	ANS:	

PTS:	1	DIF:	Easy	REF:	6.3 Radian Measure	
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding | Communication

	17.	ANS:	

PTS:	1	DIF:	Moderate	REF:	6.3 Radian Measure	
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding | Communication

	18.	ANS:	
Sketches may vary. For example:

PTS:	1	DIF:	Easy	REF:	6.3 Radian Measure	
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Communication

	19.	ANS:	
Sketches may vary. For example:

PTS:	1	DIF:	Easy	REF:	6.3 Radian Measure	
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Communication

	20.	ANS:	
Expressions may vary. For example:

PTS:	1	DIF:	Easy	REF:	6.3 Radian Measure	
LOC:	12.T1	TOP:	Trigonometry			KEY:	Conceptual Understanding

	21.	ANS:	
p, p

PTS:	1	DIF:	Moderate	REF:	6.3 Radian Measure	
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

	22.	ANS:	

PTS:	1	DIF:	Moderate	REF:	6.3 Radian Measure	
LOC:	12.T3	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding

	23.	ANS:	
Approximately  radians

PTS:	1	DIF:	Easy	REF:	6.3 Radian Measure	
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding

	24.	ANS:	
184 cm2

PTS:	1	DIF:	Moderate	REF:	6.3 Radian Measure	
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding

	25.	ANS:	
Approximately 3.9 cm

PTS:	1	DIF:	Moderate	REF:	6.3 Radian Measure	
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding

	26.	ANS:	
Approximately 5.7 cm

PTS:	1	DIF:	Difficult	REF:	6.3 Radian Measure	
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding

	27.	ANS:	
p radians/s

PTS:	1	DIF:	Moderate	REF:	6.3 Radian Measure	
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding

	28.	ANS:	

PTS:	1	DIF:	Easy	REF:	6.4 Graphing Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			KEY:	Procedural Knowledge

	29.	ANS:	
1 zero

PTS:	1	DIF:	Moderate	REF:	6.4 Graphing Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			KEY:	Conceptual Understanding

	30.	ANS:	
, , 

PTS:	1	DIF:	Moderate	REF:	6.4 Graphing Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

	31.	ANS:	
, , 

PTS:	1	DIF:	Moderate	REF:	6.4 Graphing Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

	32.	ANS:	

PTS:	1	DIF:	Easy	REF:	6.5 Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Communication	

	33.	ANS:	

PTS:	1	DIF:	Easy	REF:	6.5 Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Communication	

	34.	ANS:	

PTS:	1	DIF:	Easy	REF:	6.5 Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Communication	

	35.	ANS:	

PTS:	1	DIF:	Moderate	REF:	6.5 Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Communication	

	36.	ANS:	
range: 

PTS:	1	DIF:	Easy	REF:	6.5 Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

	37.	ANS:	
Yes.

PTS:	1	DIF:	Easy	REF:	6.5 Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			KEY:	Conceptual Understanding

	38.	ANS:	
Equations may vary. For example:
,  Z

PTS:	1	DIF:	Moderate	REF:	6.5 Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

	39.	ANS:	
•	amplitude: 4
•	period: 
•	equation of the centre line: 
•	phase shift: 

PTS:	1	DIF:	Easy	REF:	6.6 Combining Transformations of Sinusoidal Functions
LOC:	12.T4	TOP:	Trigonometry			KEY:	Procedural Knowledge

	40.	ANS:	
•	amplitude: 1
•	period: 
•	equation of the centre line: 
•	phase shift: 
•	zeros: 
•	domain: 
•	minimum value: –1
•	maximum value: 1
•	range: 

PTS:	1	DIF:	Easy	REF:	6.6 Combining Transformations of Sinusoidal Functions
LOC:	12.T4	TOP:	Trigonometry			KEY:	Procedural Knowledge

	41.	ANS:	
•	amplitude: 9
•	period: 
•	equation of the centre line: 
•	phase shift: 

PTS:	1	DIF:	Easy	REF:	6.6 Combining Transformations of Sinusoidal Functions
LOC:	12.T4	TOP:	Trigonometry			KEY:	Procedural Knowledge

	42.	ANS:	
+5

PTS:	1	DIF:	Easy	REF:	6.6 Combining Transformations of Sinusoidal Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

	43.	ANS:	
+6

PTS:	1	DIF:	Moderate	REF:	6.6 Combining Transformations of Sinusoidal Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

	44.	ANS:	
Students’ answers will vary depending on whether they use the sine or cosine function, on the phase shift they identify, and on whether they include a reflection in the x-axis. For example:

PTS:	1	DIF:	Moderate	REF:	6.6 Combining Transformations of Sinusoidal Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

	45.	ANS:	
Students’ answers will vary depending on whether they use the sine or cosine function, on the phase shift they identify, and on whether they include a reflection in the x-axis. For example:

PTS:	1	DIF:	Moderate	REF:	6.6 Combining Transformations of Sinusoidal Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

	46.	ANS:	
Students’ answers will vary depending on whether they use the sine or cosine function, on the phase shift they identify, and on whether they include a reflection in the x-axis. For example:

PTS:	1	DIF:	Moderate	REF:	6.6 Combining Transformations of Sinusoidal Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

	47.	ANS:	
Students’ answers for the phase shift may vary.
•	amplitude: 2
•	period: 
•	equation of the centre line: 
•	phase shift: 	•	maximum value: 1
•	minimum value: –3
•	domain: 
•	range: 	

PTS:	1	DIF:	Easy	REF:	6.7 Applications of Sinusoidal Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

	48.	ANS:	
Students’ answers may vary. For example:
.

PTS:	1	DIF:	Moderate	REF:	6.7 Applications of Sinusoidal Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

	49.	ANS:	
Students’ answers may vary. For example:

PTS:	1	DIF:	Moderate	REF:	6.7 Applications of Sinusoidal Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

	50.	ANS:	
42 cm

PTS:	1	DIF:	Moderate	REF:	6.7 Applications of Sinusoidal Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

	51.	ANS:	
15 cm

PTS:	1	DIF:	Moderate	REF:	6.7 Applications of Sinusoidal Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

	52.	ANS:	
10 in.

PTS:	1	DIF:	Easy	REF:	6.7 Applications of Sinusoidal Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

	53.	ANS:	
26 in.

PTS:	1	DIF:	Easy	REF:	6.7 Applications of Sinusoidal Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

	54.	ANS:	
Every  s

PTS:	1	DIF:	Moderate	REF:	6.7 Applications of Sinusoidal Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

	55.	ANS:	
The maximum fraction of the moon was seen on October 29 or 30.
The minimum fraction of the moon was seen on October 15.

PTS:	1	DIF:	Easy	REF:	6.7 Applications of Sinusoidal Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

PROBLEM

	1.	ANS:	
Sketch a diagram.
Let the length of OP be r.  
Use:       Substitute  and .

 

 

 	

 

 	

PTS:	1	DIF:	Moderate	
REF:	6.1 Trigonometric Ratios for Any Angle in Standard Position	
LOC:	12.T3	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding

	2.	ANS:	
A coterminal angle is: 
The terminal arm of this angle lies in Quadrant 3, where the tangent ratio is positive, and the sine and cosine ratios are negative. The reference angle is .
Determine the coordinates of point P(x, y) on the terminal arm where it intersects the unit circle.  

                      and

	

	

PTS:	1	DIF:	Moderate	
REF:	6.1 Trigonometric Ratios for Any Angle in Standard Position	
LOC:	12.T3	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding

	3.	ANS:	
The terminal arm of angle q lies in Quadrant 3.
The reference angle is: 
In Quadrant 3, , or 

Sketch a diagram.

The angles that are coterminal with  in the domain  are approximately:

Possible values of q are approximately:  and .

PTS:	1	DIF:	Moderate	
REF:	6.1 Trigonometric Ratios for Any Angle in Standard Position	
LOC:	12.T3	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

	4.	ANS:	

Since  is negative, the terminal arm of angle q  lies in Quadrant 2 or Quadrant 3.
The reference angle is:  

In Quadrant 2: , or 
In Quadrant 3: , or  

Sketch a diagram.

An angle that is coterminal with  in the domain  is:

An angle that is coterminal with  in the domain  is:

So, the possible measures of angle q  are , , , and 

PTS:	1	DIF:	Moderate	
REF:	6.1 Trigonometric Ratios for Any Angle in Standard Position	
LOC:	12.T3	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding

	5.	ANS:	

Since  is positive, the terminal arm of angle q  lies in Quadrant 1 or Quadrant 2.
The reference angle is: 

In Quadrant 1, 
In Quadrant 2, , or  

Sketch a diagram.

An angle that is coterminal with  in the domain  is:

An angle that is coterminal with  in the domain  is:

So, the possible measures of angle q  are approximately , , , and .

PTS:	1	DIF:	Moderate	
REF:	6.1 Trigonometric Ratios for Any Angle in Standard Position	
LOC:	12.T3	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding

	6.	ANS:	
The wheel has a diameter of 38 m, so its radius is 19 m.  
The length of the arc that subtends a central angle of 80° in a circle with radius 19 m represents the total distance the chair travels.
arc length  
	    = p
The chair travels through a distance of p metres.

PTS:	1	DIF:	Moderate	REF:	6.2 Angles in Standard Position and Arc Length
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge | Problem-Solving Skills

	7.	ANS:	
The radius of the wheel, in inches, is: 
The distance the wheel rolls, in inches, is: p
The wheel rolls p inches backward.

PTS:	1	DIF:	Moderate	REF:	6.2 Angles in Standard Position and Arc Length
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge | Problem-Solving Skills

	8.	ANS:	
The radius of the reel, in centimetres, is: 
The length, in centimetres, is: p
The length of fishing line pulled in is p centimetres.

PTS:	1	DIF:	Moderate	REF:	6.2 Angles in Standard Position and Arc Length
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge | Problem-Solving Skills

	9.	ANS:	
The terminal arm of angle q  lies in Quadrant 2.
The reference angle is: 
In Quadrant 2, , or approximately 

Sketch a diagram.

The angles that are coterminal with  in the domain  are approximately:

Possible values of q  are approximately: –10.2 and –3.9

PTS:	1	DIF:	Moderate	REF:	6.3 Radian Measure	
LOC:	12.T3	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge

	10.	ANS:	

Since  is negative, the terminal arm of angle q  lies in Quadrant 2 or Quadrant 4.
The reference angle is:  

In Quadrant 2: , or 
In Quadrant 4: , or  

Sketch a diagram.

An angle that is coterminal with  in the domain  is:

An angle that is coterminal with  in the domain  is:

So, the possible measures of angle q are , , , and 

PTS:	1	DIF:	Moderate	REF:	6.3 Radian Measure	
LOC:	12.T3	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding

	11.	ANS:	

Since  is negative, the terminal arm of angle q  lies in Quadrant 2 or Quadrant 3.
The reference angle is: 

In Quadrant 2, , or 
In Quadrant 3, , or  

Sketch a diagram.

An angle that is coterminal with 1.8235 in the domain  is:

An angle that is coterminal with 4.4597 in the domain  is:

So, the possible measures of angle q  are approximately 1.8, –4.5, 4.5, and –1.8

PTS:	1	DIF:	Moderate	REF:	6.3 Radian Measure	
LOC:	12.T3	TOP:	Trigonometry			
KEY:	Procedural Knowledge | Conceptual Understanding

	12.	ANS:	
a)	The circumference of the rink is:  m.
	In 59 s, she skates around  of the circumference of the rink. 
	
	In 59 s, the distance she skates along the arc is approximately 17.7 m.

b)	The central angle subtended by the arc in part a is: , or approximately 1.0413 radians
	The radius of the rink is: , or 17 m
	The straight-line distance is the length of the third side of a triangle with two sides equal to 17 m and a contained angle of 1.0413. 
	Use the Cosine Law to determine the length of the side: 
	
	The child skates approximately 16.9 m.

PTS:	1	DIF:	Moderate	REF:	6.3 Radian Measure	
LOC:	12.T1 | 12.T3			TOP:	Trigonometry		
KEY:	Conceptual Understanding | Problem-Solving Skills | Procedural Knowledge

	13.	ANS:	

The area, in square metres, of the larger circle is: 
The area, in square metres, of sector AOB is  of the area of the larger circle.
 
The area, in square metres, of the smaller circle is: 
The area of sector COD is  of the area of the smaller circle.

		

The area of the shaded region is  square metres.

PTS:	1	DIF:	Difficult	REF:	6.3 Radian Measure	
LOC:	12.T1	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Problem-Solving Skills | Procedural Knowledge

	14.	ANS:	
Create a table of values for angles x in the domain , then determine coterminal angles in the domain –2pp.

x	0						
coterminal angles in –2pp	, 0, 	, 	, 	, 	, 	, 	
	1			0			

x	p						
coterminal angles in pp	, p	, 	, 	, 	, 	, 	
	–1			0			

PTS:	1	DIF:	Moderate	REF:	6.4 Graphing Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge | Communication

	15.	ANS:	
Create a table of values for angles x in the domain , then determine coterminal angles in the domain –pp.

x	0						
coterminal angles in –pp	0, 	, 	, 	, 	, 	, 	
	0			undefined			

x	p						
coterminal angles in pp	, p, 	, 	, 	, 	, 	, 	
	0			undefined			

PTS:	1	DIF:	Moderate	REF:	6.4 Graphing Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge | Communication

	16.	ANS:	

Sample Answer:
, so:
- The zeros of  correspond to the vertical asymptotes of the graph of , and vice versa.
- The points on  with y-coordinates of 1 or –1 are common to both graphs.

PTS:	1	DIF:	Moderate	REF:	6.4 Graphing Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge | Communication

	17.	ANS:	
The graph of  is the image after the graph of  has been translated 3 units down. 

The amplitude is 1.
The period is p.
There are no zeros.
There are no asymptotes.
The domain is .
The range is .

PTS:	1	DIF:	Easy	REF:	6.5 Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge | Communication

	18.	ANS:	
The graph of  is the image after the graph of  has been stretched vertically by a factor of . 

The amplitude is 3.
The period is p.
The zeros are  Z.
The domain is .
The range is .

PTS:	1	DIF:	Moderate	REF:	6.5 Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge | Communication

	19.	ANS:	
The graph of  is the image after the graph of  has been horizontally stretched by a factor of  and reflected in the y-axis. 

The amplitude is 1.
The period is p.
The zeros are  Z.
There are no asymptotes.
The domain is .
The range is .

PTS:	1	DIF:	Moderate	REF:	6.5 Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge | Communication

	20.	ANS:	
The graph of  is the image after the graph of  has been translated  units horizontally. 

There is no amplitude.
The period is p.
The zeros are  Z.
The asymptotes are 
The domain is .
The range is .

PTS:	1	DIF:	Moderate	REF:	6.5 Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge | Communication

	21.	ANS:	
a)	The graph of  was stretched or compressed horizontally by a factor of .
	If , the graph was also reflected in the y-axis.

b)	If , the graph of  was translated  units to the right. 
	If , the graph of  was translated  units to the left.

PTS:	1	DIF:	Moderate	REF:	6.5 Trigonometric Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Communication

	22.	ANS:	
Students’ answers will vary depending on the phase shift they identify, and whether they include a reflection in the x-axis. For example:
The equation has the form .
The maximum y-value is 4 and the minimum y-value is –2, so the amplitude is  and .
The distance between adjacent maximum points is , so the period is , and , or .
The equation of the centre line is , so .
At , the graph of  reaches a maximum, so look for a point near the y-axis where a maximum occurs. 
A maximum occurs at , so the phase shift is , and .
So, , , , and 
Substitute into .
An equation is: +1

PTS:	1	DIF:	Moderate	REF:	6.6 Combining Transformations of Sinusoidal Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge | Communication

	23.	ANS:	
Compare  to :
, so the graph of  is stretched vertically by a factor of 2, and the amplitude is 2.
, so the graph is stretched horizontally by a factor of , and the period is , or .
, so the graph is translated  units left, and the phase shift is .
, so the graph is translated 2 units down, and the centre line has equation: 
The domain is: 
The maximum value of the function is: 
The minimum value of the function is: 
So, the range is: 

PTS:	1	DIF:	Moderate	REF:	6.6 Combining Transformations of Sinusoidal Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge | Communication

	24.	ANS:	
Compare  to :
, , , and 
The graph of  is stretched vertically by a factor of 3, compressed horizontally by a factor of , and then translated 1 unit up.

, so the amplitude is 3.
, so the period is , or .
, so the phase shift is 0.
, so the equation of the centre line is . 
The domain is .
The maximum y-value is 3 units above the centre line and the minimum y-value is 3 units below the centre line, so the range is .

PTS:	1	DIF:	Moderate	REF:	6.6 Combining Transformations of Sinusoidal Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge | Communication

	25.	ANS:	
Compare  to :
, , , and 
The graph of  is stretched vertically by a factor of 4, compressed horizontally by a factor of , then translated  units right and 1 unit up.

, so the amplitude is 4.
, so the period is , or .
, so the phase shift is .
, so the equation of the centre line is . 
The domain is .
The maximum y-value is 4 units above the centre line and the minimum y-value is 4 units below the centre line, so the range is .

PTS:	1	DIF:	Moderate	REF:	6.6 Combining Transformations of Sinusoidal Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge | Communication

	26.	ANS:	
The equation has the form .
The amplitude is 4, so .
The period is , so , or .
The equation of the centre line is , so .
The phase shift is , so  and the cosine function has a maximum at .
So, , , , and .
Substitute into 
An equation is: –2
The graph of –2 is the graph of  after it is stretched vertically by a factor of 4, compressed horizontally by a factor of , then translated  units right and 2 units down.
 –2

PTS:	1	DIF:	Moderate	REF:	6.6 Combining Transformations of Sinusoidal Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge | Problem-Solving Skills | Communication

	27.	ANS:	
Write  in the form . 

Compare to : 
, , , and 
The graph of  is stretched vertically by a factor of 2, compressed horizontally by a factor of , reflected in the x-axis, and then translated  units left and 1 unit up.

, so the amplitude is 2.
, so the period is , or .
, so the phase shift is .
, so the equation of the centre line is . 
The domain is .
The maximum y-value is 2 units above the centre line and the minimum y-value is 2 units below the centre line, so the range is .

PTS:	1	DIF:	Difficult	REF:	6.6 Combining Transformations of Sinusoidal Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge | Communication

	28.	ANS:	
Equations may vary. For example:

Graph the data. Let f represent the visible fraction of the moon, and t represent the day.

The beginning of the period is a maximum point, so use a cosine function to model the data: 

From the table and the graph:
The first maximum point has approximate coordinates (5, 1), and the first minimum point has approximate coordinates (19, 0), so the equation of the centre line is approximately: , or ; so 
The amplitude is approximately: ; so 
The period is approximately 30 days, so , or 
The phase shift is the t-coordinate of the first maximum point, so 
A function that approximates the data is: 

PTS:	1	DIF:	Moderate	REF:	6.7 Applications of Sinusoidal Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge | Problem-Solving Skills | Communication

	29.	ANS:	
At  s, the mark is the same height above the table as the axle. The mark will move up when the fan is turned on. So, use a sine function.
An equation that models the height of the mark has the form: 
The mark is 24 cm from the axle, so the amplitude is 24, and .
The axle is 30 cm above the table, so the equation of the centre line is , and .
The fan completes one rotation in  s, so the period is  s.

Since the mark starts at the same height as the axle and moves up, the phase shift is 0 and .
Substitute , p, , and  into .
An equation that represents the height of the mark is: 

PTS:	1	DIF:	Moderate	REF:	6.7 Applications of Sinusoidal Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge | Problem-Solving Skills | Communication

	30.	ANS:	
The diameter of the tire is the maximum value of the function: 55 cm.
The circumference of the tire, in centimetres, is:

There are  cm in 1 km. So, the circumference of the tire, in kilometres, is: 
The period of the function is 0.05 s, so the car travels forward a distance equivalent to one tire circumference, or  km, every 0.05 s.  
So, in 1 s, it travels:  km
And, in 1 h, itr travels:  km  
The car’s average speed is approximately 124 km/h.

PTS:	1	DIF:	Difficult	REF:	6.7 Applications of Sinusoidal Functions
LOC:	12.T4	TOP:	Trigonometry			
KEY:	Conceptual Understanding | Procedural Knowledge | Problem-Solving Skills