MATHEMATICS 17: Additional Exercises on Circular and Trigonometric Functions

and Identities

I. Establish the identities.

1.1− cos θ

sin θ=

sin θ

1 + cos θ

2.tan θ

1− cot θ+

cot θ

1 − tan θ= 1 + tan θ + cot θ

3. sec θ csc θ − 2cos θ csc θ + cot θ = tan θ

4.sec θ + tan θ

cos θ − tan θ − sec θ= − csc θ

5. sin3 θ − cos3 θ = sin θ(1 + sin θ cos θ) − cos θ(1 + sin θ cos θ)

6.sin3θ

sin θ−

cos3θ

cos θ= 2

7.tan θ

2+ cot θ

2

cot θ

2− tan θ

2

= sec θ

8. sin4θ = 4cos θ(sin θ − 2sin3 θ)

9. csc2θ + cot 2θ − cot θ = 0

10. sin2 θ

4+ cos θ

2= cos2 θ

4

11.1− cos8x

8= sin2 2x cos2 2x

12.sin x + sin 3x + sin 5x

cos x + cos3x + cos5x= tan3x

II. Do as indicated.

1. Identify the amplitude, period, phase shift, and vertical displacement and sketch one cycleof the graph of each of the following functions.

a. f (x) =1

2cos

x

2+

π

6

− 2b. g(x) = 3 sin(π − 4x) −2

3c. h(x) = 1+2 cos

πx

2− π

2. Given:tanα = 3

4,α does not lie in Quadrant I, sin β = −

5

13cotβ > 0. Find:

a. sec(π2− β ) [Answer: −13

5]

b. sin(α − β ) [Answer: 16

65]

c. cot(α− β ) [Answer: 63

16]

d. csc(3π4

+ α) [Answer: −5√ 

2]

3. Find all the circular functions of  θ if sec θ = −5

3and tan θ < 0.

[Answer: cos θ = −3

5, sin θ = 4

5, tan θ = −

4

3, cot θ = −

3

4, csc θ = 5

4]

4. Find all trigonometric functions of α if the terminal side of α (in standard position) passesthrough

a. (3, 0)[Answer: cosα = 1, sin α = 0, tan α = 0, cot α = undefined, secα = 1, csc α =undefined]

b. (−7, 1)

[Answer: cosα = −7√ 2

10, sinα =

√ 2

10, tanα = −

1

7, cotα = −7, secα = −

5√ 2

7, cscα =

5√ 

2]

5. Find all the trigonometric functions of  θ if csc θ = −3 and θ does not lie in Quadrant IV.

[Answer: sin θ = −1

3, cos θ = −

2√ 2

3, tan θ =

√ 2

2, cot θ = 2

√ 

2, sec θ = −3√ 2

4]

6. Give the values of the trigonometric functions for the following angles. (You may use theconcept of reference angles.)

a. 120◦ (Let θ = 120◦.)

[Answer: sin θ =√ 3

2, cos θ = −

1

2, tan θ = −

√ 

3, cot θ = −

√ 3

3, sec θ = −2, csc θ = 2

√ 3

3]

b. −390◦ (Let θ = −390◦.)

[Answer: sin θ = −1

2, cos θ =

√ 3

2, tan θ = −

√ 3

3, cot θ = −

√ 

3, sec θ = −2√ 3

3, csc θ = 2]

7. Express cos 71π

11as a function of a real number α where 0 < α < π

2.

[Answer: cos 5π

11or sin π

22]

8. Express the following as a function of a positive angle less than 45 ◦ or of a real number

less thanπ

4.

a. sin 47π

9[Answer: − sin 2π

9]

b. cos1021◦ [Answer: sin 31◦]

9. Evaluate the following.

a. cos157◦30 (Recall: 1◦ = 60 ) [Answer: −

√ 

2+√ 2

2]

b.tan

π

18− tan

11π

36

1 + tanπ

18tan

11π

36

[Answer: −1]

c. sin7.5◦ cos37.5◦ [Answer:√ 2−14

]

d. sin125◦ sin20◦ + cos20◦ cos125◦ [Answer:√ 2−

√ 6

4]

e.sin 4π

15− sin 6π

15

sin 2π

15cos π

15− cos 2π

15sin π

15

. [Answer: −1]

10. Show that the following are not identities.

a. cos x + cos3x = 2cos2x [Answer: Take x = π

2]

b. sin4θ =√ 

1 − cos2 4θ [Answer: Take θ = π

3]

11. Solve the following problems.

a. If  f (θ) = cos

π − θ

3

, find [f (2π) − f (0)]. [Answer: 0]

b. If sin2θ =1

4, find the value of (cos θ + sin θ)8. [Answer: 625

256]

“Study as if you have not reached your goal - hold it as if you were afraid of losing what you have.” - Confucius

/mmrobles 09.19.09/