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Question: How do you simplify \({\sin ^{ - 1}}\left( {\cos \left( {\dfrac{{7\pi }}{6}} \right)} \right)\)...

How do you simplify sin⁡−1(cos⁡(7π6)){\sin ^{ - 1}}\left( {\cos \left( {\dfrac{{7\pi }}{6}} \right)} \right)

Explanation

Solution

Here first we write 7π6\dfrac{{7\pi }}{6} as (π+π6)\left( {\pi + \dfrac{\pi }{6}} \right) and convert cosine into sine function and we will get the value of the given inverse function.

Formula used:
The trigonometric identities used here are:
First one is
cos⁡ (π+a) = −cos⁡ a\cos \,\left( {\pi + a} \right)\, = \, - \cos \,a
Second one is
−cos⁡ a = −sin⁡ (π2−a)- \cos \,a\, = \, - \sin \,\left( {\dfrac{\pi }{2} - a} \right)
Third one is
− sin⁡ a = sin⁡ (−a)- \,\sin \,a\, = \,\sin \,\left( { - a} \right)

Complete step by step solution:
Here we can write cos⁡(7π6)\cos \left( {\dfrac{{7\pi }}{6}} \right)as cos⁡ (π + π6)\cos \,\left( {\pi \, + \,\dfrac{\pi }{6}} \right)
Using the trigonometric identity we can write
cos⁡ (π + π6) = −cos⁡ (π6)\cos \,\left( {\pi \, + \,\dfrac{\pi }{6}} \right)\, = \, - \cos \,\left( {\dfrac{\pi }{6}} \right)
Now, using the formula−cos⁡ a = −sin⁡ (π2−a) - \cos \,a\, = \, - \sin \,\left( {\dfrac{\pi }{2} - a} \right), we can write
−cos⁡ (π6) = −sin⁡ (π2−π6)- \cos \,\left( {\dfrac{\pi }{6}} \right)\, = \, - \sin \,\left( {\dfrac{\pi }{2} - \dfrac{\pi }{6}} \right)
This can be simplified to − sin⁡ (π3) - \,\sin \,\left( {\dfrac{\pi }{3}} \right)
Now, using− sin⁡ a = sin⁡ (−a) - \,\sin \,a\, = \,\sin \,\left( { - a} \right),
we can write − sin⁡ (π3) = sin⁡ (−π3) - \,\sin \,\left( {\dfrac{\pi }{3}} \right)\, = \,\sin \,\left( { - \dfrac{\pi }{3}} \right)
Hence, it can be written as sin⁡−1(cos⁡(7π6)) = −π3{\sin ^{ - 1}}\left( {\cos \left( {\dfrac{{7\pi }}{6}} \right)} \right)\, = \, - \dfrac{\pi }{3}
Therefore the value of the given function is −π3 - \dfrac{\pi }{3}

Note:
First we should know the different properties of the trigonometric function in order to solve the question easily. It is also important for us to keep in mind the quadrant in which all functions are positive or negative.
Trigonometric equations are those equations which contain trigonometric functions i.e. Sine, Cosine, Tangent, Cosecant, Secant and Cotangent.
A function of an angle expressed as the ratio of two of the sides of a right angle that angle is called trigonometric functions.
The sine, cosine, tangent, cotangent, secant and consent are the trigonometric functions.

There are three main functions in trigonometry i.e. Sine, Cosine and Tangent. There are certain trigonometric identities which can be stated as below:
sin⁡x = 1cos⁡ec x\sin x\, = \,\dfrac{1}{{\cos ec\,x}}
cos⁡x = 1sec⁡ x\cos x\, = \,\dfrac{1}{{\sec \,x}}
tan⁡x =1cot⁡x\tan x\, = \dfrac{1}{{\cot x}}
The sine and the cosecant are the inverse of each other. The cosine and the secant are the inverse of each other. The tangent and the cotangent are inverse of each other. They all are related to each other in the special formulas which are called trigonometric identities.