Question

find the following without using a calculator. cos ( sin ^ { - 1 } \frac { sqrt { 2 } } { 2 } ) = \frac { sqrt { 2 } } { {? } }

Explanation:

Step1: Find the value of \(\sin^{-1}\frac{\sqrt{2}}{2}\)

Let \(\theta=\sin^{-1}\frac{\sqrt{2}}{2}\). By the definition of inverse - sine function, \(\sin\theta=\frac{\sqrt{2}}{2}\) and \(\theta\in[-\frac{\pi}{2},\frac{\pi}{2}]\). We know that \(\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}\), so \(\theta = \frac{\pi}{4}\).

Step2: Calculate \(\cos(\sin^{-1}\frac{\sqrt{2}}{2})\)

Since \(\sin^{-1}\frac{\sqrt{2}}{2}=\frac{\pi}{4}\), then \(\cos(\sin^{-1}\frac{\sqrt{2}}{2})=\cos\frac{\pi}{4}\). And we know that \(\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}\).

Answer:

\(\frac{\sqrt{2}}{2}\)