Question

How do you evaluate $\arccos \left( 0 \right)$ without a calculator?

Answer 1

Here, we have to evaluate the value of the inverse of the trigonometric function. We will use the trigonometric ratio of cosine of an angle to evaluate the trigonometric function. The inverse functions are also called arc functions since the value of a trigonometric function produces the length of the arc from that value

Formula Used:
Trigonometric Ratio: $\cos \dfrac{\pi }{2} = 0$

Complete step by step solution:
It is given that $\arccos \left( 0 \right)$.
We know that the Trigonometric Ratio $\cos \dfrac{\pi }{2} = 0$.
Taking inverse cosine function on both sides, we get
$\Rightarrow {\cos ^{ - 1}}\left( {\cos \dfrac{\pi }{2}} \right) = {\cos ^{ - 1}}\left( 0 \right)$
Now using the identity ${\cos ^{ - 1}}\left( {\cos x} \right) = x$ in above equation, we get
$\Rightarrow \dfrac{\pi }{2} = {\cos ^{ - 1}}\left( 0 \right)$
$\Rightarrow \arccos \left( 0 \right) = \dfrac{\pi }{2}$

Therefore, the value of $\arccos \left( 0 \right)$ is $\dfrac{\pi }{2}$.

Additional Information:
We know that Trigonometric Equation is defined as an equation involving trigonometric ratios. Trigonometric identity is an equation which is always true for all the variables. We should know that we have many trigonometric identities that are related to all the other trigonometric equations. Trigonometric Ratios of a Particular angle are the ratios of the sides of a right angled triangle with respect to any of its acute angle. Trigonometric Ratios are used to find the relationships between the sides of a right angle triangle.

Note:
We need to keep in mind that the inverse trigonometric function is used to find the missing angles in a right angled triangle whereas the trigonometric function is used to find the missing sides in a right angled triangle. The range of the arc cosine of the angle lies between $- \dfrac{\pi }{2}$ radians and $\dfrac{\pi }{2}$ radians. The maximum of the cosine of the angle is 1 and at $\dfrac{\pi }{2}$ radians. The basic angles used in solving trigonometric problems are in degrees. The trigonometric angles can also be denoted in Radians. .