Question

Find the evaluation of \[{\cos ^{ - 1}}\left( { - \dfrac{{\sqrt 2 }}{2}} \right)\].

Answer 1

Hint:The inverse trigonometric functions also called arcus functions, ant trigonometric functions or cyclometric functions are the inverse functions of the trigonometric functions. Specifically they are the inverses of the sine, cosine, tangent, cotangent, secant and cosecant functions and are used to obtain an angle from any of the angle’s trigonometric ratios.

Complete step by step solution:
According to the question we need to evaluate \[{\cos ^{ - 1}}\left( { - \dfrac{{\sqrt 2 }}{2}} \right)\]and the first step that has to be done is to assume a variable equivalent to this.
Hence we have,
\[A = {\cos ^{ - 1}}\left( { - \dfrac{{\sqrt 2 }}{2}} \right)\]
Now we will multiply \[\sqrt 2 \]in both numerator and denominator and rationalize it.
Hence, after multiplying we have
\[ \Rightarrow A = {\cos ^{ - 1}}\left( { - \dfrac{1}{{\sqrt 2 }}} \right)\]
We know that,
\[{\cos ^{ - 1}}\left( { - x} \right) = \pi - {\cos ^{ - 1}}x\]
Hence we can solve this further as shown below:
\[ \Rightarrow A = \pi - {\cos ^{ - 1}}\left( {\cos \left( {\dfrac{\pi }{4}} \right)} \right).....\]
As we know that\[\cos \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }}\],
Hence this can be further simplified as,
\[
\Rightarrow A = \pi - \frac{\pi }{4} \\
\Rightarrow A = \frac{{3\pi }}{4} \\
\]
Since we can obtain this by having an LCM that is lowest common multiple which is 4 and when subtracted then we obtain the evaluation of \[{\cos ^{ - 1}}\left( { - \dfrac{{\sqrt 2 }}{2}}
\right)\]as\[\dfrac{{3\pi }}{4}\].

Note: Since none of the six trigonometric functions are one to one, they must be restricted in order to have inverse functions. Therefore, the ranges of the inverse functions are proper subsets of the domains of the domains of the original functions. Each of the trigonometric functions is periodic in the real part of its argument, running through all its values twice in each interval of \[2\pi \]. Cosine and secant begin their period at \[2\pi k\], finish it at \[2\pi k + \pi \], and then reverse themselves over \[2\pi k + \pi \] to \[2\pi k + 2\pi \]. This periodicity is reflected in the general inverses where “k” is some integer.