Pre-Calculus Math 40s - Trigonometry I...

Trigonometry I

Trigonometry I Standards Test Practice Exam - ANSWERS 0 www.math40s.com Copyright © 2006, Barry Mabillard. www.math40s.com

1. The minimum and the maximum of a trigonometric function are shown in the diagram. a) Write a cosine equation for the function

Trigonometry I Standards Test Practice Exam - ANSWERS 1 www.math40s.com

First determine the parameters a, b, c, & d.

( )7 - -3max - min 10a= = = = 5

2 2 2

*The horizontal distance between the two points is 8 units, so a full cycle (the period) is 16 units.

π π π2 2b= = =

P 16 8

c = 5 (that is where the cosine pattern begins) min+max -3+7 4

d = = = = 22 2 2

The equation is ( )⎡ ⎤⎢ ⎥⎣ ⎦

y = 5cos x-5 +28π

b) Determine the value of the y – intercept, correct to three decimal places Find the y-intercept by setting x = 0, then solving for y.

( )

( )

π

π

π

⎡ ⎤⎢ ⎥⎣ ⎦⎡ ⎤⎢ ⎥⎣ ⎦⎡ ⎤⎢ ⎥⎣ ⎦

y = 5cos x -5 +28

y = 5cos 0 -5 +28

-5y = 5cos +2

8y =0.087

Alternatively, graph in your calculator and type 2nd → Trace → Value → x = 0

2. Convert 1118π to degrees

ππ

o11 180× =

18o110

3. Find the exact value of sinθ if 1cos7

θ = , and the terminal arm is in Quadrant I.

First draw in the triangle, using the fact adj

cosθ=hyp


Use Pythagoras to find the unknown side:

( ) ( )2 2 2

22 2

2

2

1 7

1 76

6

a b c

b

bb

b

+ =

+ =

+ =

=

= ±

*Reject the negative case since we know the terminal arm is in quadrant I.

It follows that 6si nθ=

7 4. What is the period in the function ( ) ( )3sin 5 2f x x= −

The b –value is 5

The period is π2

P = =b

25π

5. Given the point (-3, -4), determine the exact values of cosθ and sinθ Draw the information you know, then use Pythagoras to solve for the unknown side:

( ) ( )

2 2 2

2 2 2

2

2

3 4

9 1625

5

a b c

c

cc

c

+ =

− + − =

+ =

==

-3

cosθ=5

-4sinθ=

5

6. The length of arc swept out by an angle θ is 50 cm. If the radius of the circle is 18 cm, determine the measure of θ in radians.

Use the arc length formula s=θr

( )50 cm=θ 18 cm

θ=2.78 rad

7. Determine a value for x that would make the function ( ) 3cscf x x= undefined in the domain [0,2 )π

Recall that 1

cscx =sinx

, so look for values of x that make si nx = 0

A possible answer is 0 rad or π rad.

2π rad may not be used because it is not included in the allowed domain for the question.

8. Determine the value of in the function ( )10f ( ) ( )4sin 2 3f x xπ= − +⎡ ⎤⎣ ⎦

( ) ( )( ) [ ]( ) ( )( )

π

π

⎡ ⎤⎣ ⎦f 10 =4sin 10-2 +3

f 10 =4sin 8 +3

f 10 =4 0 +3

f 10 =3

9. The graph of 36cos 14

y x π⎛ ⎞= +⎜ ⎟⎝ ⎠

+ is illustrated below.

Determine the exact values of g, h, m & n. From the function, a = 6. The b – value is 1.

π3phase shift =

4 (shifted left)

d = 1.


g: 34π

−

h: 1 + 6 = 7 m: (Half a period past point g)

3 3 44 4 4 4π π π ππ− + = − + =

n: 1 – 6 = -5

10. If the measure of the central angle is 4π , determine the measures of the other two

angles within the triangle. All angles in a triangle add

up to 180°. (or π). In an isosceles triangle, the two angles marked by x are equivalent. It follows that:

Therefore, the measure

of the angles is πo

067.5 × =180

38π

45 1802 135

67.5

o o

o

o

x xx

x

+ + =

=

=

11. Determine the number of radians between the hour hand and the minute hand at 7:00.

7:00 is exactly 712

of the

way around the circle. If 2π is the whole circle, then

π 72 × =

1276π


12. Determine the value of 7cos4π⎛ ⎞−⎜ ⎟

⎝ ⎠

π7

-4

is a co-terminal angle with π4

π⎛ ⎞⎜ ⎟⎝ ⎠

2cos =

4 2

13. The circle is drawn below, along with the line 2 2 1x y+ =3

2y = .

Determine the coordinates of the two intersection points.

The points on the unit circle that have a

y – coordinate of 32

are located at

π π2 and

3 3

The coordinates at π3

are ⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

1 3,

2 2

The coordinates at π2

3 are

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

1 3- ,

2 2

14. The graph of ( ) 6sinf x x= + d touches the x-axis once (but does not pass through) on the interval 0 2x π≤ ≤ . A possible value for d is: There are two possibilities: The graph could touch at the top or at the bottom.

The d-value here is -6 The d-value is +6


15. a) Graph cosy x= b) Graph 1cosy x−=


c) State the domain of ( )1f x−

{ }| -1 1,x x x R≤ ≤ ∈ 16. The maximum point on a trigonometric graph is at the point (-4, 6), and the minimum point is at (2, -2). If the graph is of the form ( )cosy a b x c d= + +⎡ ⎤⎣ ⎦ , then determine possible values for each of the parameters.

( )6 - -2max - min 8a= = = = 4

2 2 2

* The period is 12 units π π π2 2

b= = =P 12 6

c = 4 ( )6+ -2max+min 4

d = = = = 22 2 2

The equation is ( )π⎡ ⎤⎢ ⎥⎣ ⎦

y = 4cos x+4 +26

17. An angle of 15° is equivalent to _______ radians. (exact value)

πoo15 × =

180 12π

18. The exact value of 1 5cos cos6π− ⎛ ⎞

⎜ ⎟⎝ ⎠

is

You can do this using degree mode in your calculator:

( )

o

-1 o

cos150 =-0.8660

cos -0.8660 = 150 or 56π

19. Write the general equation of a vertical asymptote in the graph of cscy x=

Rewrite csc x as 1

sinx. Vertical asymptotes occur when sin x = 0

A vertical asymptote occurs when x = 0, π, 2π, and so on. The general solution would be { }| ,x x k k Iπ= ∈ 20. If csc 2θ = and tan 0θ < , determine the value of cosθ Since cscθ is positive, and tanθ is negative, the terminal arm must be in quadrant II

Recall that hyp 2

cscθ= =opp 1

Use Pythagoras to determine the unknown side.

2 2

2 2 2

2

2

1 21 43

3

a b caaa

a

2+ =

+ =

+ =

=

= −

Use a negative since the base of the triangle lies on the negative x-axis.

It follows that 3

cosθ=-2


21. Given the function ( ) ( )2sin 22

f x xπ⎡ ⎤= −⎢ ⎥⎣ ⎦

a) Sketch the graph b) Sketch ( )y f x= 22. If the coordinates of a point ( )P θ on the unit circle is (a, b), then the coordinates of

the point are 180oP θ⎡ ⎤+⎣ ⎦ This question should be visualized on the unit circle:

If you look at a point exactly 180° away from point P(a, b), you can see that both coordinates change sign. The coordinates of ⎡ ⎤⎣ ⎦

oP θ+180 are (-a, -b)


23. State the period of the graph of cscy θ=

The period of is 2π y = cscθ

24. Convert 35π to degrees. Express answer to one decimal place.

ππ

o3 180× =

5o108

25. Given the function ( ) tanf x x=

a) Sketch on the domain ( )y f x= ,2 2π π⎛ ⎞−⎜ ⎟

⎝ ⎠

b) State the domain of ( )f x

| ,2

x x n nπ π⎧ ⎫≠ + ∈⎨ ⎬⎩ ⎭

I

c) Sketch the graph of ( )1f x−


26. A floating ball in a lake goes up and down with the tide. At 1 second, the ball has a minimum height of 5 cm below surface level. At 3 seconds, the ball has a maximum height of 5 cm above surface level. a) Sketch a graph for the first 4 seconds of motion b) Write an equation for the function

a = 5 π π π2 2

b= = =P 4 2

c = 2 d = 0

( ) ( )⎛ ⎞⎜ ⎟⎝ ⎠

h t = 5sin t - 22π

OR ( ) π⎛ ⎞⎜ ⎟⎝ ⎠

h t =-5sin t2

27. If the product of cos x and sin x is negative, then which quadrant is the angle in?

This is true in the quadrants where x & y have different signs. The answer is quadrants II & IV

28. The value of 1 1sin2

− ⎛ ⎞⎜ ⎟⎝ ⎠

is

In your calculator (using degree mode)

evaluate sin-1(0.5) = 30° (or 6π

)

29. The exact value of sin cos2π⎛ ⎞

⎜ ⎟⎝ ⎠

is

( )π⎛ ⎞⎜ ⎟⎝ ⎠

sin cos = sin 0 =2

0


30. If the coordinates of a point ( )P θ on the unit circle are 1 3,2 2

⎛ ⎞−⎜⎜

⎝ ⎠⎟⎟ , then the

coordinates of the point are 180oP θ⎡ ⎤+⎣ ⎦

Based on the coordinates, the angle is 300°. If the point is rotated 180°, the new angle is 120°.

The coordinates are ⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

1 3- ,

2 2

31. A tire rolls 3π metres while turning 240°. Determine the area of the wheel.

Use the arc length formula. (First convert the angle to radians!)

ππ

ππ

π π

⎛ ⎞⎜ ⎟⎝ ⎠

s=θr

43 = r

34 r

3 =3

9 =4 r9

r =4

Now use the formula for the area of a circle:

π

π ⎛ ⎞⎜ ⎟⎝ ⎠

2

2

A= r

9A=

42A=15.9 m


Multiple Choice Answers:

1. Given 5csc4

θ = , the exact value of tanθ is

a) 54

First rewrite the equation as 4

sinθ=5

Without any further information, the angle could be in quadrant I or the angle could be in QII Both cases will give a positive value for sinθ Use Pythagoras to solve for the unknown sides.

Based on these triangles, 4

tanθ=±3

The answer is d.

b) 35

c) 34

±

d) 4±3


2. A tire has a radius of 5π m. The tire is rolled and travels a total distance of 28π m.

By the time the tire stops, it has rolled through an angle of

Use the arc length formula s =θr The arc length s is π28 m

The radius is π5

m

ππ

ππ

π π

⎛ ⎞⎜ ⎟⎝ ⎠

28 =θ5

θ28 =

5140 =θθ=140

This means the angle is 140 radians. Since this is not a solution, convert to degrees.

π

00180

140 rad× =8021.41

The answer is d.

a) 28π b) 140°

c) 28 rad5

d) 8021.41° 3. Given the trigonometric function ( ) cosf x = x , the statement which is true is

a) ( ) ( )f x f= − x

−2π −3π2

−π −π2

π2

π 3π2

2π

-1

-0.5

0.5

1

Based on the above graph, it can be seen that a reflection about the y – axis will result in the exact same graph. The answer is b.

b) ( ) ( )f x = f -x

c) ( ) ( )1f x f x−=

d) ( ) ( )f x f= − −x



0

0

4. If , a true statement is 090 180θ< <a) 0 00 9θ< <

The angle is in quadrant II. We know that and

0sin90 =00sin180 =1

Therefore, 0<sinθ<1 The answer is c.

b) cos tanθ θ≤ c) 0 < sinθ < 1d) 0 cos 1θ< <

5. Given the function ( ) ( )12cos 2f x = x , and the transformation ( ) ( )14

g x f x= ,

then the amplitude of is ( )g xa) 2

Start with ( ) ( )1g x = f x

4.

Replace ( )f x with the actual function

( ) ( )⎡ ⎤⎣ ⎦1

g x = 12cos 2x4

Now multiply the fraction into the brackets ( ) ( )g x =3cos 2x

The answer is b.

b) 3 c) 4 d) 12

6. The function ( ) 2secf x = x

)

has a range of

a) ( ) 2, 2−

From the diagram, it can be seen that the graph exists beneath -2, and above 2. The answer is b.

b) ( , 2] [2,−∞ − ∪ ∞

c) 1 1,2 2

⎡ ⎤−⎢ ⎥⎣ ⎦

d) ( ) ,−∞ ∞ 7. A wheel turns through an angle of 16 radians. This angle measurement in degrees is

a) 0

8π

π π

0 0180 288016 rad× =

The answer is d. b) 018

π

c) 8 0π

d) 02800

π

8. A sine function has a range of [-6, 2] and a period of 4. A trigonometric equation with these properties is

a) 8sin 22

y π θ⎛ ⎞= +⎜ ⎟⎝ ⎠


b) ( )4sin 4 2y θ= −

c) ⎛ ⎞⎜ ⎟⎝ ⎠πy = 4sin θ - 22

d) 24sin 2y θπ⎛ ⎞= −⎜ ⎟⎝ ⎠

( )

π π π

2- -6max-min 8a= = = =4

2 2 22 2

b= = =P 4 2

c=0min+max -6+2 -4

d= = = =-22 2 2

The equation is π⎛ ⎞⎜ ⎟⎝ ⎠

y=4sin θ -22

The answer is c.

9. The angle 243π is co-terminal to an angle of

a) 0˚ π π 0

0 0 0 0 0

24=8 =1440

31440 -360 -360 -360 -360 =0

b) 23π

0

The answer is a. c) 56π

d) 73π

10. The exact value of 7csc4π⎛ ⎞

⎜ ⎟⎝ ⎠

is ππ

⎛ ⎞⎜ ⎟ ⎛ ⎞⎝ ⎠

⎜ ⎟⎝ ⎠

7 1 1csc = = =-

74 2 2sin -4 2

2

Rationalize the denominator:

2 2 -2 2- × = =- 2

22 2

The answer is a.

a) - 2

b) 22

−

c) 22

d) 2

11. Given that 4cos5

θ = and 3sin5

θ = − , then the terminal arm is located in quadrant

a) I A positive cosine means the terminal arm is located in quadrant I or IV A negative sine means the terminal arm is located in quadrant III or IV There is overlap in IV. The answer is d.

b) II c) III d) IV 12. The period of the function g(x) is 8. If ( ) ( ) ( )0 12, g 4 6, and g 8 12g = = = , then the

value of is ( )12gDraw in the values to see what is happening, then project the graph to the x-value of 12. The value of ( )g 12 is 6.

The answer is b.

a) 0 b) 6 c) 12 d) 18


13. An angle of 32π on the unit circle has coordinates of

a) 1 3,2 2

The coordinates can be read off the unit circle, and are (0, -1) The answer is c. b) 2 2,

2 2−

c) (0, -1) d) (-1, 0) 14. If sec 0θ > and sin 0θ < , then θ terminates in quadrant a) I b) II A positive secant means the terminal

arm is located in quadrant I or IV A negative sine means the terminal arm is located in quadrant III or IV There is overlap in IV. The answer is d.

c) III d) IV

15. The period and phase shift for the trigonometric equation 1 sin 42 3

y πθ⎛ ⎞= −⎜ ⎟⎝ ⎠

are

a) πperiod = 4 ; phase shift = left3


b) πperiod = 4 ; phase shift = right3

c) πperiod = ; phase shift = left2 3π

d) π πperiod = ; phase shift = right2 3

The period is π π π2 2

= =b 4 2

.

The phase shift is 3π

units right.

The answer is d.

16. The equation of an asymptote on the graph of cscy x= is

a) 4

x π=

The answer is c.

b) 2

x π=

c) x = π

d) 32

x π=

17. If , then x terminates in quadrants tan sin 0x x <a) II or III b) II or IV tanxsinx<0 occurs when

tanx and sinx have different signs. (Recall the product of a positive and negative is negative) They have different signs in quadrants II (tanx is negative, sin x is positive) and quadrant III (tanx is positive, sinx is negative.) The answer is a.

c) I or IV d) III or IV

18. The value of 1 2sin2

− ⎛ ⎞⎜⎜⎝ ⎠

⎟⎟ is:

a) π4

When π

θ=4

, the value of sinθ is 22

Alternatively, type in ⎛ ⎞⎜⎜⎝ ⎠

-1 2sin

2

b) 54π

⎟⎟ into

your calculator in degree mode to get back 450

The answer is a.

c) 43π

d) 116π


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