Exam (elaborations) Inverse trigonometry

Subject:- Mathematics
Topic:- Inverse Trigonometry MCQ
Level:-All entrances

𝜋 1
1. sin [ − 𝑠𝑖𝑛−1 (− )] is equal to
3 2

1 1 1
a) b) c) d) 1
2 3 4

A. The answer is d) 1
𝜋 1 𝜋 1 𝜋
soln:- 𝑠𝑖𝑛 [ − 𝑠𝑖𝑛−1 (− )] = sin [ + 𝑠𝑖𝑛−1 ( )] = sin = 1
3 2 3 2 2




1
2. If 𝑐𝑜𝑠 −1 = 𝜃, then 𝑡𝑎𝑛𝜃 is equal to
𝑥

1
a) b) √𝑥 2 − 1
√𝑥 2 −1

c) √1 − 𝑥 2 d) √𝑥 2 + 1

A. The answer is b) √𝑥 2 − 1
1
soln:- Now, 𝑐𝑜𝑠 −1 = 𝜃
𝑥

1
√1− 2
−1 𝑥
⇨ 𝑡𝑎𝑛 [ 1 ]=𝜃
𝑥



∴ 𝑡𝑎𝑛𝜃 = tan⁡[𝑡𝑎𝑛−1 √𝑥 2 − 1 = √𝑥 2 − 1


5𝜋2
3. If (𝑡𝑎𝑛−1 𝑥)2 + (𝑐𝑜𝑡 −1 𝑥)2 = , then 𝑥 is equal to
8

a) −1 b) 1 c) 0 d) 2

, A. The answer is a) −1

soln:- Let 𝑡𝑎𝑛−1 𝑥 = 𝑦
𝜋
Then,⁡𝑐𝑜𝑡 −1 𝑥 = − 𝑦
2

𝜋 2 5𝜋2
∴ ⁡⁡⁡𝑦² + ( − 𝑦) =
2 8

⇨ 16𝑦 2 − 8𝜋𝑦 − 3𝜋 2 = 0

⇨ (4𝑦 − 3𝜋)(4𝑦 + 𝜋) = 0
𝜋
⇨ 𝑦 = 𝑡𝑎𝑛−1 𝑥 = −
4

3𝜋 𝜋
[ ⁡rejected, since it is greater than ]
4 4

𝜋
∴ ⁡⁡⁡⁡⁡𝑥 = tan (− ) = −1
4


1
4. Let −1 ≤ 𝑥 ≤ 1. If cos(𝑠𝑖𝑛−1 𝑥) = , then how many values does tan⁡(𝑐𝑜𝑠 −1 𝑥) assume?
2

a) One b) Two c) Four d) Infinite

A. The answer is b) two
1
soln:- cos(𝑠𝑖𝑛−1 𝑥) =
2

1 1
⇨ cos(𝑐𝑜𝑠 −1 √1 − 𝑥 2 ) = ⇨ √1 − 𝑥 2 =
2 2

1 √3
⇨ ⁡⁡⁡⁡⁡⁡⁡⁡1 − 𝑥² = ⇨⁡𝑥 = ±
4 2

Hence, tan⁡(𝑐𝑜𝑠 −1 𝑥) have two values


2𝑥 1−𝑥 2 2𝑥 𝜋
5. If 3𝑠𝑖𝑛−1 ( ) − 4𝑐𝑜𝑠 −1 (1+𝑥 2) + 2𝑡𝑎𝑛−1 (1−𝑥 2) = 3 , then 𝑥 is equal to?
1+𝑥 2

a) √3 b) 1/√3 c) 1 d) −1
1
A. The answer is b)
√3