Solveeit Logo
Login

Question

Question: The equation \(2 \cos ^ { - 1 } x + \sin ^ { - 1 } x = \frac { 11 \pi } { 6 }\) has...

The equation 2cos⁡−1x+sin⁡−1x=11π62 \cos ^ { - 1 } x + \sin ^ { - 1 } x = \frac { 11 \pi } { 6 } has

A

No solution

B

Only one solution

C

Two solutions

D

Three solutions

Answer

No solution

Explanation

Solution

Given equation is 2cos⁡−1x+sin⁡−1x=11π62 \cos ^ { - 1 } x + \sin ^ { - 1 } x = \frac { 11 \pi } { 6 }

⇒ cos⁡−1x+(cos⁡−1x+sin⁡−1x)=11π6\cos ^ { - 1 } x + \left( \cos ^ { - 1 } x + \sin ^ { - 1 } x \right) = \frac { 11 \pi } { 6 }

⇒ cos⁡−1x=11π6−π2\cos ^ { - 1 } x = \frac { 11 \pi } { 6 } - \frac { \pi } { 2 }

⇒cos⁡−1x=4π3\Rightarrow \cos ^ { - 1 } x = \frac { 4 \pi } { 3 }, which is not possible as cos⁡−1x∈[0,π]\cos ^ { - 1 } x \in [ 0 , \pi ] .