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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/