Question Set 1 1. Convert the following angles to radians: (a) 310° (b) −110° (c) 118.38° (nearest hundredth)

11

2. Convert the following angles to degrees: (a) 5𝜋

6 (b) −

11𝜋

2 (c) 3.2 (nearest degree)

3. Find each missing value: provide any angles in both radians (nearest hundredth) and degrees (nearest degree)

SE Solve multi-step problems based on the relationship 𝜃 =𝑎

𝑟 (angle conversion is not considered a step)

Question Set 2

SOLUTIONS

RTD’s Math 30-1

TRIG WORKSHOP

To “Question Set” items

(a) (b)

(c)

Given that the two circles have the same radius,

determine the measure of 𝜃 in the second circle.

(d) The minute hand on the clock shown

measures 18 cm. Assuming it

represents the radius of the circle,

determine the length of the arc

formed by the hour and minute hand

at 10:10am.

4. State one positive and one negative co-terminal angle (in both degrees and radians) for each of the following:

5. For each angle in #5, state an expression for all co-terminal angles. (in both degrees and radians)

6. State the principal angle for each of the following:

7. For each angle in standard position, state the quadrant which the terminal arm lies in, and the measure of the

reference angle. (Draw a diagram)

(a) 𝜃 = 261° (b) 5𝜋

7 (c) 920° (d)

20𝜋

3

8. An angle in standard position 𝜃 passes through a point (-4, -5). Determine the exact value of all six trigonometric

ratios of 𝜃, and the measure of 𝜃. (Correct to the nearest degree and hundredth of a radian)

(a) (b)

(a) (b)

Question Set 2

9. Given that 0 ≤ 𝜃 < 2𝜋, 𝑐𝑜𝑠𝜃 =5

√61 and 𝑡𝑎𝑛𝜃 < 0, determine:

(a) The exact value of 𝑐𝑠𝑐𝜃 and 𝑐𝑜𝑡𝜃.

(b) The approximate value of 𝜃. (Correct to the nearest degree and hundredth of a radian)

10. An angle in standard position passes through a point 𝑃(−𝑏, 2𝑏). Determine:

(a) The exact value of 𝑐𝑜𝑠𝜃.

(b) The approximate value of 𝜃. (Correct to the nearest degree and hundredth of a radian)

11. If 𝑡𝑎𝑛𝜃 = −4

5, determine the largest possible value of 𝜃 on 0 ≤ 𝜃 < 2𝜋. (nearest hundredth)

12. If 𝑠𝑒𝑐𝜃 = −3

2 and 𝑡𝑎𝑛𝜃 > 0, determine (a) the exact value of 𝑠𝑖𝑛𝜃, and (b) The approximate value of 𝜃 in degrees

(nearest whole number) and radians (nearest hundredth).

13. An angle in standard position 𝜃 passes through a point 𝑃(𝑎, 𝑏). If the measure of 𝜃 is 2𝜋

3 and the distance from 𝑃 to

the origin is 5 units, determine the exact values of 𝑎 and 𝑏.

14. An angle in standard position 𝜃 passes through a point 𝑃(0, 4). State an expression for all angles co-terminal to 𝜃.

(in both degrees and radians) (nearest degree / hundredth of a radian)

Question Set 3

15. Determine 16. For each of the following unit circle diagrams, determine the missing coordinate (as an exact value), and determine the

value of 𝜃, in degrees and (for (a) only exact, in terms of 𝜋) radians.

𝜃 𝜃

(𝑥,1

2)

(𝑥,12

13)

17. An 18. On the

19. In the diagram below, the reference angle is 𝛼 =𝜋

6. State the value of 𝜃 (degrees and exact radians).

𝛼

20. A

21. Determine the exact value of each trigonometric ratio. Draw the terminal arm on each unit circle.

22. Determine quadrant of the terminal arm of the angle in standard position, and state the approximate value of each trig ratio:

(a) sin7𝜋

5 (b) sec(−295°) (c) cot 17.5

23. State the solutions to each equation, for 0 ≤ 𝜃 < 2𝜋: (Where possible, exact solutions in terms of 𝜋)

(a) sin 𝜃 =1

2 (b) cos 𝜃 = −

√3

2 (c) csc 𝜃 = √2

24. An angle 𝜃 has tan 𝜃 = √3 and cos 𝜃 < 0. Determine the value of 𝜃, in degrees and radians.

25. An angle 𝜃 has sin 𝜃 = −1

3 and sec 𝜃 > 0. Determine the value of 𝜃, correct to the

26. If the point 𝑃(0.2, 𝑘) lies on a circle with a radius of 1, then the exact value of 𝑘 can be what two values?

27. Determine the exact value of sin (−𝜋

6) + 𝑐𝑜𝑠 (

7𝜋

4).

28. P

𝒂 = 𝟕, 𝒃 = 𝟏𝟐

29.

Question Set 3 30. State the period, amplitude, domain, and range for the graphs of each of the following functions.

(a) 𝑦 = 3sin (4𝑥) (b) 𝑦 = −2 cos (1

4𝑥) (c) 𝑦 = tan (3𝑥)

31. Determine sine equation for each of the following graphs:

32. State the period, amplitude, domain, and range for the graphs of each of the following functions.

(a) 𝑦 = −1.25 sin (3

2𝑥) − 1.5 (b) 𝑦 = 200 cos (

𝑥

50) + 200 (c) 𝑦 = 20.4 sin(2.8𝑥) − 10.4

33. State the period, amplitude, domain, range, and horizontal phase shift for the graphs of each of the following

functions.

(b) 𝑦 = 4 sin[3(𝑥 − 75°)] − 5 (b) 𝑦 = 5 cos[1

8(𝑥 +

𝜋

6)] + 1 (c) 𝑦 = cos(4𝑥 − 𝜋) − 1

(b) (a)

State period to nearest hundredth

34. Determine both a sine and cosine equation for each of the following graphs:

35. The

Provide a complete explanation of how the characteristics of the graph of a trigonometric function relate to the conditions in a contextual situation. SE

(a) (b)

Determine the complete equation, finding all 4 parameters, for a sinusoidal curve given the graph, the characteristics, or a real-world situation SE

36. The graph shows how the height of a bicycle pedal changes as the bike is pedalled at a constant speed.

Question Set 4a

37. Given the identity 𝑠𝑖𝑛𝑥

𝑡𝑎𝑛𝑥= 𝑐𝑜𝑠𝑥,

(a) Verify using the angles 10° and 𝜋

6 (b) Verify graphically. Sketch resulting graphs here

(b) Prove algebraically

(c) State any non-permissible values

38. Simplify each of the following trig expressions to 𝑠𝑖𝑛𝜃, 𝑐𝑜𝑠𝜃, or 1.

(a) 𝑠𝑖𝑛𝜃𝑐𝑜𝑡𝜃 (b) 𝑡𝑎𝑛𝜃𝑠𝑒𝑐𝜃𝑐𝑜𝑠2𝜃 (c) 𝑐𝑜𝑡𝜃𝑐𝑜𝑠𝜃𝑐𝑠𝑐𝜃𝑡𝑎𝑛2𝜃

Left Side Right Side

(a) Determine a cosine equation to model the height of the pedal, ℎ, as a

function of time in seconds, 𝑡. (In the form 𝑦 = 𝑎cos 𝑏[(𝑥 − 𝑐)] + 𝑑.)

(b) Use your equation to predict how many seconds (nearest tenth) the pedal is

above 40cm in the first 10 s.

(c) Which equation parameter(s) (a, b, c, or d) would change if the bicycle were

pedalled at a greater constant speed?

39. Simplify into a single trigonometric expression.

(a) 𝑐𝑜𝑡𝑥𝑡𝑎𝑛𝑥

𝑐𝑠𝑐𝑥 (b)

𝑠𝑖𝑛𝑥

1−𝑐𝑜𝑠2𝑥 (c)

𝑠𝑖𝑛𝑥 𝑐𝑠𝑐2𝑥

𝑠𝑒𝑐𝑥 (d)

1−𝑠𝑖𝑛2𝑥

𝑐𝑜𝑡2𝑥∗ 𝑐𝑠𝑐𝑥

40. Consider the equation 𝑠𝑒𝑐𝑥 + 𝑡𝑎𝑛𝑥 =𝑐𝑜𝑠𝑥

1−𝑠𝑖𝑛𝑥.

(a) State the non-permissible values, in radians.

41. ANS: A

42. ANS: A

(b) Verify that this equation is true using an angle 𝑥 =𝜋

4.

(c) PROVE that this is an identity.

43.

Question Set 4b #43

1. Use the appropriate formula to evaluate: cos(90 30 )

Use cos(𝛼 − 𝛽) = 𝑐𝑜𝑠𝛼𝑐𝑜𝑠𝛽 + 𝑠𝑖𝑛𝛼𝑠𝑖𝑛𝛽

cos(90° − 30°) = 𝑐𝑜𝑠90°𝑐𝑜𝑠30° + 𝑠𝑖𝑛90°𝑠𝑖𝑛30°

= (0) (√3

2) + (1) (

1

2)

= 0 +1

2 =

𝟏

𝟐

2. Use the Pythagorean Identity to express the identity cos(2𝛼) = 𝑐𝑜𝑠2𝛼 − 𝑠𝑖𝑛2𝛼 entirely in terms of 𝑠𝑖𝑛𝛼. Pyth. Identity is 𝑐𝑜𝑠2𝛼 + 𝑠𝑖𝑛2𝛼 = 1

So we can re-arrange to get 𝑐𝑜𝑠2𝛼 = 1 − 𝑠𝑖𝑛2𝛼

So now cos(2𝛼) = 𝑐𝑜𝑠2𝛼 − 𝑠𝑖𝑛2𝛼

cos(2𝑎) = (1 − 𝑠𝑖𝑛2𝛼) − 𝑠𝑖𝑛2𝛼 𝐜𝐨𝐬(𝟐𝜶) = 𝟏 − 𝟐𝒔𝒊𝒏𝟐𝜶

3. Express as a single trig function:

(a) cos 50 cos 20 sin50 sin20

Use cos(𝛼 − 𝛽) = 𝑐𝑜𝑠𝛼𝑐𝑜𝑠𝛽 + 𝑠𝑖𝑛𝛼𝑠𝑖𝑛𝛽 as what we’re given “fits that pattern”!

𝛼 = 50°, 𝛽 = 20° … So we have cos(50° − 20°) =𝐜𝐨𝐬(𝟑𝟎°)

(b) 𝑠𝑖𝑛𝜋

5𝑐𝑜𝑠

𝜋

8+ 𝑐𝑜𝑠

𝜋

5𝑠𝑖𝑛

𝜋

8

Use sin(𝛼 + 𝛽) = 𝑠𝑖𝑛𝛼𝑐𝑜𝑠𝛽 + 𝑐𝑜𝑠𝛼𝑠𝑖𝑛𝛽

𝛼 =𝜋

5, 𝛽 =

𝜋

8 … So we have sin (

𝜋

5+

𝜋

8) =𝐬𝐢𝐧 (

𝟏𝟑𝝅

𝟒𝟎)

(c) 𝑡𝑎𝑛20°+𝑡𝑎𝑛15°

1−𝑡𝑎𝑛20°𝑡𝑎𝑛15°

Use tan(𝛼 + 𝛽) =𝑡𝑎𝑛𝛼+𝑡𝑎𝑛𝛽

1−𝑡𝑎𝑛𝛼𝑡𝑎𝑛𝛽

𝛼 = 20°, 𝛽 = 15° … So we have tan(20° + 15°) =𝐭𝐚𝐧(𝟑𝟓°)

4. Simplify sin (𝜋

2− 𝜃)

= sin (𝜋

2) 𝑐𝑜𝑠𝜃 − cos (

𝜋

2) 𝑠𝑖𝑛𝜃

= (1)(𝑐𝑜𝑠𝜃) − (0)(𝑠𝑖𝑛𝜃) = 𝒄𝒐𝒔𝜽

6. Use the appropriate double-angle formula to evaluate:

(a) sin(2 330 )

Use sin (2𝛼) = 2𝑠𝑖𝑛𝛼𝑐𝑜𝑠𝛽, with 𝛼 = 330°

= 2𝑠𝑖𝑛330°𝑐𝑜𝑠330°

= 2(−1

2)(

√3

2) = −

√𝟑

𝟐

(b) 2𝑡𝑎𝑛

𝜋

12

1−𝑡𝑎𝑛2 𝜋

12

Use tan(2𝛼) =2𝑡𝑎𝑛𝛼

1−𝑡𝑎𝑛2𝛼, with 𝛼 =

𝜋

12

= tan (2 ∗𝜋

12) = tan (

𝜋

6) =

√𝟑

𝟑

This is 1 − 𝑠𝑖𝑛2𝛼

5. Express as a single trigonometric function: 2𝑐𝑜𝑠2 5𝜋

12− 1

Use cos(2𝛼) = 2𝑐𝑜𝑠2𝛼 − 1, with 𝛼 =5𝜋

12

= cos (2 ∗5𝜋

12) = cos (

10𝜋

12) = 𝐜𝐨𝐬 (

𝟓𝝅

𝟔)

www.rtdmath.com

www.rtdmath.com

7. Express as a single trig function:

(a) 2 sin60 cos 60

Use sin(2𝛼) = 2𝑠𝑖𝑛𝛼 𝑐𝑜𝑠𝛼, with 𝛼 = 60°

= sin (2 ∗ 60°)

= 𝐬𝐢𝐧 (𝟏𝟐𝟎°)

8. Simplify the function 𝑐𝑜𝑠2𝑥 𝑐𝑜𝑠𝑥 + 𝑠𝑖𝑛2𝑥 𝑠𝑖𝑛𝑥

Use cos(𝛼 − 𝛽) = 𝑐𝑜𝑠𝛼𝑐𝑜𝑠𝛽 + 𝑠𝑖𝑛𝛼𝑠𝑖𝑛𝛽, with 𝛼 = "2𝑥" and 𝛽 = "𝑥"

= cos (2𝑥 − 𝑥)

= 𝒄𝒐𝒔𝒙

9. Given 4

cos5

A and 7

sin25

B , where 3

22

A

and 3

2B

, find the exact value of sin( )A B .

10. Find the exact value of sin12

First, change the angle to degrees so it’s easier to come up with two unit circle angles that add or subtract to it. = 𝑠𝑖𝑛15°

Next, come up with two unit circle angles that add or subtract to 15°.

= sin (45° − 30°) Finally, use sin(𝛼 − 𝛽) = 𝑠𝑖𝑛𝛼𝑐𝑜𝑠𝛽 − 𝑐𝑜𝑠𝛼𝑠𝑖𝑛𝛽

sin (45° − 30°) = 𝑠𝑖𝑛45°𝑐𝑜𝑠30° − 𝑐𝑜𝑠45°𝑠𝑖𝑛30°

= (√2

2) (

√3

2) − (

√2

2) (

1

2)

=√6

4−

√2

4 =

√𝟔−√𝟐

𝟒

= cos (2 ∗𝜋

6)

= 𝐜𝐨𝐬(𝝅

𝟑)

(b) 2 2cos sin6 6

Use cos(2𝛼) = 𝑐𝑜𝑠2 ∝ −𝑠𝑖𝑛2𝛼, with 𝛼 =

𝜋

6

11. Find the exact value of 𝑡𝑎𝑛17𝜋

12

Convert to degrees… = tan (255°) < Express in terms of unit circle angles

= tan (225° + 30°) < This is just one option

tan(𝛼 + 𝛽) =𝑡𝑎𝑛𝛼 + 𝑡𝑎𝑛𝛽

1 − 𝑡𝑎𝑛𝛼𝑡𝑎𝑛𝛽

Here, 𝛼 = 225° and 𝛽 = 30°

=𝑡𝑎𝑛225° + 𝑡𝑎𝑛30°

1 − 𝑡𝑎𝑛225°𝑡𝑎𝑛30°

=1+√3/3

1−(1)(√3/3) =

3+√3

33−√3

3

=3+√3

3∗

3

3−√3 =

3+√3

3−√3 ∗

3+√3

3+√3

=𝟏𝟐+𝟔√𝟑

𝟔 = √𝟑 + 𝟐

Prove the identity 2𝑡𝑎𝑛𝑥

1−𝑡𝑎𝑛2𝑥=

sin (2𝑥)

𝑐𝑜𝑠2𝑥−𝑠𝑖𝑛2𝑥

(state any variable restrictions)

44. Given that 𝑡𝑎𝑛𝑥 =3

4, where 180° < 𝑥 < 270°, the exact value of cos (𝑥 − 30°) is:

A. 3√3+4

10

B. −3√3−4

10

C. 3+4√3

10

D. −3−4√3

10

LS RS

=

2𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑥

𝑐𝑜𝑠2𝑥𝑐𝑜𝑠2𝑥

−𝑠𝑖𝑛2𝑥𝑐𝑜𝑠2𝑥

=

2𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑥

𝑐𝑜𝑠2𝑥 − 𝑠𝑖𝑛2𝑥𝑐𝑜𝑠2𝑥

=2𝑠𝑖𝑛𝑥

𝑐𝑜𝑠𝑥∗

𝑐𝑜𝑠2𝑥

𝑐𝑜𝑠2𝑥 − 𝑠𝑖𝑛2𝑥

=2𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑥

𝑐𝑜𝑠2𝑥 − 𝑠𝑖𝑛2𝑥

Rationalize denominator

=2𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑥

𝑐𝑜𝑠2𝑥 − 𝑠𝑖𝑛2𝑥

45. Simplify each of the following to a single numerical value:

(a) 𝑐𝑜𝑡2𝑥 − 𝑐𝑠𝑐2𝑥

(b) 𝑠𝑒𝑐2𝑥 − 𝑡𝑎𝑛2𝑥

(c) 𝑠𝑖𝑛𝑥 −𝑡𝑎𝑛𝑥

𝑠𝑒𝑐𝑥

(d) 1

7𝑐𝑜𝑠2𝑥 +

1

7𝑠𝑖𝑛2𝑥

46. (Added in class)

I.E.E.#3

I.E.E.#1 Simplify Consider the solution to the equation 𝑠𝑖𝑛2𝑥 − 𝑐𝑜𝑠𝑥 = 0 ; 0 ≤ 𝑥 < 2𝜋

Solve graphically: Solve algebraically:

I.E.E.#2 Determine a general solution to the equation 2𝑐𝑜𝑠2𝑥 − 𝑐𝑜𝑠𝑥 − 1 = 0

TRIGONOMETRIC EQUATIONS

Answer: “B”

I.E.E.#3 ANS: “B”

I.E.E.#4

=

2𝑡𝑎𝑛𝑥

1 + 𝑡𝑎𝑛2𝑥