Question

find the following without using a calculator.
cos(cos⁻¹ 0.3) = ?

Explanation:

Step1: Recall inverse cosine property

The cosine function and its inverse function, \( \cos^{-1}(x) \) (also written as \( \arccos(x) \)), are inverse operations. For a function \( f(x) \) and its inverse \( f^{-1}(x) \), the composition \( f(f^{-1}(x)) = x \) for all \( x \) in the domain of \( f^{-1}(x) \) (and the range of \( f(x) \)). Here, \( f(x)=\cos(x) \) and \( f^{-1}(x)=\cos^{-1}(x) \), and \( 0.3 \) is within the domain of \( \cos^{-1}(x) \) (since the domain of \( \cos^{-1}(x) \) is \( [-1, 1] \) and \( 0.3\in[-1, 1] \)).

Step2: Apply the inverse function property

Using the property \( \cos(\cos^{-1}(x))=x \) for \( x\in[-1, 1] \), when \( x = 0.3 \), we have \( \cos(\cos^{-1}(0.3))=0.3 \).

Answer:

\( 0.3 \)