Question

How do you do inverse cosine?

Answer 1

Hint: Trigonometric functions have inverse functions called inverse trigonometric functions. Inverse cosine is an inverse trigonometric function. It is the inverse of the cosine function.

Complete step-by-step answer:
We are familiar with the fact which is given as $\cos \theta =\dfrac{\text{Adjacent} \text{side}}{\text{Hypotenuse}}$ where $\theta $ is an angle of a right-angled triangle.
If we are asked to find the angle $\theta $ when the adjacent side and the hypotenuse are given, what we have to do is to find the value of inverse cosine of the quotient $\dfrac{\text{Adjacent} \text{side}}{\text{Hypotenuse}}.$
That is, we have to find ${{\cos }^{-1}}\dfrac{\text{Adjacent} \text{side}}{\text{Hypotenuse}}.$
This is just a case. But this is the way we do inverse cosine.
It would be easy if we know how to calculate the cosine values. Because if we know that, then the only part remaining is to find the angle for which the cosine value is the given value.
Suppose that we are given with the angle $\theta =\dfrac{\pi }{6}.$ It is known that the cosine of $\dfrac{\pi }{6}$ is $\dfrac{\sqrt{3}}{2}.$
That is, $\cos \dfrac{\pi }{6}=\dfrac{\sqrt{3}}{2}.$
So, from this we can see the inverse cosine value of $\dfrac{\sqrt{3}}{2}$ is $\dfrac{\pi }{6}.$
Thus, we can write it as ${{\cos }^{-1}}\dfrac{\sqrt{3}}{2}=\dfrac{\pi }{6}.$
Similarly, we know that $\cos \dfrac{\pi }{4}=\dfrac{1}{\sqrt{2}}.$
Now this implies ${{\cos }^{-1}}\dfrac{1}{\sqrt{2}}=\dfrac{\pi }{4}.$
So, let us suppose that if we are asked to find the inverse cosine of $\dfrac{1}{2}.$ That is, we are asked to find the value of ${{\cos }^{-1}}\dfrac{1}{2}.$
Now, we would be able to find the inverse cosine function if we know that $\cos \dfrac{\pi }{3}=\dfrac{1}{2}.$
We will get ${{\cos }^{-1}}\dfrac{1}{2}=\dfrac{\pi }{3}.$
Similarly, ${{\cos }^{-1}}0=\dfrac{\pi }{2},{{\cos }^{-1}}1=0,$ et cetera.

Note: The domain of the cosine function is $\mathbb{R}$ and range of the cosine function is $\left[ -1,1 \right].$ The domain of the inverse cosine function is $\left[ -1,1 \right]$ and the range of the inverse cosine function is $\left[ 0,\pi \right].$ We can calculate the inverse trigonometric function values if we know the corresponding trigonometric function values.