MATHEMATICS 60
Exercises on Inverse Trigonometric Functions
I. Determine if the following statements are true or false.
1. The graph of y = Arccot x intersects the line with equation y = −1.
2. For all real numbers x ∈ [−1, 1], Arcsin (−x) = − Arcsin x.
√
3. The equation sin cos−1 x = 1 − x2 is an identity.
4. If x is positive, then the value of sin−1 cos (Arctan x) lies on the interval 0, π2 .
II. Choose the letter of the correct answer, if it is among the choices listed.
π
?
1. What is the value of Arcsin sec 10
A.
π
10
2π
5
B.
2. What is the value of cos
1
A. − 18
1
B.
2 sec
√
22
6
C.
21π
10
π
D. − 10
C.
2
3
D.
−1 (−9)
?
4
9
3. What is the value of cos−1 (cos 4)?
B. 4 − π
A. 4
C. 2π − 4
D. π + 4
III. Solve for x in terms of y.
1. y = sin−1 (x + 2)
3. 2y − tan−1 (2x) = 0
2. y = 4 Arccos x
4. 2 cos x = y, where x ∈ [0, π]
IV. Evaluate the following.
1. cos−1 (0) − csc−1 −
√ 2 3
3
√ √ 2. sin−1 − 22 · cot−1 − 3
h
√ i
√
√ 3. cos Arctan 3 + Arcsec − 2 + Arcsin − 23
π
4. cos−1 1 − 2 sin2 14
h
i
5. Arccot cot − π7
6. sin−1 cos 19π
9
h
7. cos 2 cos−1 −
5
13
i
8. sec Arccos 35 + Arcsin 1
1
h
9. sin sin−1
2
3
− cos−1
1
3
i
15
10. tan Arcsec 25
7 + Arctan 8
h
√ i
11. cos tan−1 (2) − csc−1 − 2
12. Arccos x + Arccos (−x), where x ∈ [−1, 1]
V. If θ = Arctan − 43 , find the exact value of sin 2θ and sin
θ
2
.
VI. Determine the solution set of the following equations.
1. Arccos cos(−3) − Arcsin x = π2 + 3
√ 2. Arctan − 33 + 4 Arccot x = Arcsec (−2)
cos−1 x − cot−1 (−1) = csc−1 csc 4π
3
√ √
4. 2 Arcsec 3 x − 4 − Arccot 0 = Arccos − 23
3.
1
2
√
√ x2 + 1 + tan−1 − 3 − 2 cos−1 1 = 0
√
6. Arcsin 1 − 2 Arcsec x − 2 = Arctan tan 5π
6
5.
1
2
sec−1 −
7. sin−1 x − cos−1 x =
π
6
8. Arcsin x + Arccos (1 − x) = 0
9. cos−1 (1) + 2 sin−1 x = tan−1
VII. Prove the following.
1. 2 tan−1 13 − tan−1 − 17 =
√ x
1−x2
π
4
2. Arctan 2 + Arccot (−3) = 5π
4
7
= Arccos − 54
3. Arccos − 53 + Arcsin 25
4. tan−1 12 + tan−1 51 = π4 − tan−1 18
5. sin−1 x + cos−1 x =
π
2
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“For even the very wise cannot see all ends.”
- Gandalf the Grey
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