Question

Write the principal value of ${\cos ^{ - 1}}\left[ {\cos \left( {680^\circ } \right)} \right]$.

Answer 1

In the given question, we have been given a trigonometric expression. We have to solve for the given value. We have to find the principal value. This means that the value of the answer must lie in the range of the given function. We are going to solve it using the periodicity of the trigonometric functions. The given answer must lie in the range of the standard value.

Formula Used:
We are going to use the formula of periodicity of cosine function, which is:
$\cos \left( {2n\pi - \theta } \right) = \cos \theta$

Complete step by step solution:
We have to find the principal value of ${\cos ^{ - 1}}\left[ {\cos \left( {680^\circ } \right)} \right]$.
Now, the principal value of ${\cos ^{ - 1}}$ lies in the range of $\left[ {0,\pi } \right]$ or $\left[ {0^\circ ,180^\circ } \right]$.
But, the value of the angle is more than the upper limit, so we are going to use the formula of periodicity of cosine function, which is:
$\cos \left( {2n\pi - \theta } \right) = \cos \theta$
Hence, $\cos \left( {680^\circ } \right) = \cos \left( {2\pi - 40^\circ } \right) = \cos \left( {40^\circ } \right)$
So, the principal value of ${\cos ^{ - 1}}\left[ {\cos \left( {680^\circ } \right)} \right] = {\cos ^{ - 1}}\left[ {\cos \left( {40^\circ } \right)} \right]$ is $40^\circ$.

Additional Information:
In the given question, we applied the concept of periodicity of the cosine function, so it is necessary that we know the periodicity of each trigonometric function. Periodicity of sine, cosine, cosecant, and secant is $2\pi$ while the periodicity of tangent and cotangent is $\pi$.

Note: In the given question, we found the inverse of the given trigonometric function. The argument of the trigonometric function was the same trigonometric function. The argument of this trigonometric function was an angle. So, at the first look, it would seem that the value is simply the angle itself. And it would have been if it was just asked to write the value of the expression. But it was given to write the principal value, and so we did all the calculations to get the value in the range.