How do you evaluate \[\arccos \left( 1 \right)\] without a calculator?

Hint: In this question, we have a trigonometric inverse function. Trigonometric inverse function is also called arc function. To solve the trigonometric inverse function we assume the angle \[\theta \] which is equal to that trigonometric inverse function. Then we find the value of \[\theta \].

Complete step by step answer:
In this question, we used the word trigonometric inverse function. We have the following inverse trigonometric functions,
Arcsine function: it is the inverse function of sine. It is denoted as \[{\sin ^{ - 1}}\].
Arccosine function: it is the inverse function of cosine. It is denoted as \[{\cos ^{ - 1}}\].
Arctangent function: it is the inverse function of tangent. It is denoted as\[{\tan ^{ - 1}}\].
Arccotangent function: it is the inverse function of cotangent. It is denoted as\[{\cot ^{ - 1}}\].
Arcsecant function: it is the inverse function of secant. It is denoted as \[{\sec ^{ - 1}}\].
Arccosecant function: it is the inverse function of cosecant. It is denoted as \[\cos e{c^{ - 1}}\].
Now, we come to the question. The data is given as below.
\[\arccos \left( 1 \right)\]
We can write the above trigonometric function as below.
\[ \Rightarrow \arccos \left( 1 \right) = {\cos ^{ - 1}}\left( 1 \right)\]
We know that\[\cos 0 = 1\], and then put the value of \[1\] in above.
Then,
\[ \Rightarrow \arccos \left( 1 \right) = {\cos ^{ - 1}}\left( {\cos 0} \right)\]
We know that\[{\cos ^{ - 1}}\left( {\cos \theta } \right) = \theta \].
Then, \[{\cos ^{ - 1}}\left( {\cos 0} \right) = 0\]. Put these values in above.
Hence,
\[\therefore \arccos \left( 1 \right) = 0^\circ \]

Therefore, the value of \[\arccos \left( 1 \right)\]is \[0\] degree.

Note:
As we know that the trigonometric inverse function is defined as the inverse function of trigonometric identities like sin, cos, tan, cosec, sec and cot. The trigonometric inverse function is also called cyclomatic function, anti-trigonometric function and arc function. The trigonometric inverse function is used to find the angle of any trigonometric ratio. The trigonometric inverse function is applicable for right angle triangles.