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Question: Evaluate the following: \({\cos ^{ - 1}}\left( {\cos 4} \right)\)?...

Evaluate the following: cos⁡−1(cos⁡4){\cos ^{ - 1}}\left( {\cos 4} \right)?

Explanation

Solution

This type of trigonometric problem can be easily solved if we know or have the formula to expand and change the equation in such a way that the formula gets matched so that we can easily solve it. Here we will use the formulacos⁡−1(cos⁡θ)=θ{\cos ^{ - 1}}\left( {\cos \theta } \right) = \theta . And its limit will be from(0⩽θ⩽π)\left( {0 \leqslant \theta \leqslant \pi } \right).

Formula used:
The trigonometric formula used here is:
cos⁡−1(cos⁡θ)=θ{\cos ^{ - 1}}\left( {\cos \theta } \right) = \theta ; If (0⩽θ⩽π)\left( {0 \leqslant \theta \leqslant \pi } \right)
cos⁡(2π−θ)=cos⁡θ\cos \left( {2\pi - \theta } \right) = \cos \theta
Here,
θ\theta , will be the angle made between them.

Complete step by step solution:
As we know the formula which is
cos⁡−1(cos⁡θ)=θ{\cos ^{ - 1}}\left( {\cos \theta } \right) = \theta ; If (0⩽θ⩽π)\left( {0 \leqslant \theta \leqslant \pi } \right)
And we also know that
cos⁡(2π−θ)=cos⁡θ\cos \left( {2\pi - \theta } \right) = \cos \theta
Therefore, from the above formula we have
⇒cos⁡−1(cos⁡4)\Rightarrow {\cos ^{ - 1}}\left( {\cos 4} \right)
And it can be written as by using the formula
\Rightarrow {\cos ^{ - 1}}\left\\{ {\cos \left( {2\pi - 4} \right)} \right\\}
And hence it can be written as, as we know cos⁡(2π−θ)=cos⁡θ\cos \left( {2\pi - \theta } \right) = \cos \theta
Therefore, it will be
⇒2π−4 [0⩽(2π−4)⩽π]\Rightarrow 2\pi - 4{\text{ }}\left[ {0 \leqslant \left( {2\pi - 4} \right) \leqslant \pi } \right]

Therefore, it will be cos⁡−1(cos⁡4)=⇒2π−4{\cos ^{ - 1}}\left( {\cos 4} \right) = \Rightarrow 2\pi - 4

Additional information:
As we have seen that without formula and identities we can find the values of such functions. To find the appropriate formula at one click would come through the experience gained and the more basic formulas we have memorized. Also, we should always try to use sin and cos form because most of the formulas are based on this.

Note:
To solve this type of problem we must have to know or remember the formula used in the question to simplify the expression at ease. As we can see in this question if we know cos⁡−1(cos⁡θ)=θ{\cos ^{ - 1}}\left( {\cos \theta } \right) = \theta and cos⁡(2π−θ)=cos⁡θ\cos \left( {2\pi - \theta } \right) = \cos \theta then nothing in the question is left to solve. It can be easily solved then. So we should always try to learn all the formulas related to trigonometric to solve the problem without any error and easily.