Question

Evaluate sin(sin^-1(3/4)+cos^-1(1/4))without using a calculator.

Answer 1

Let : \(sin^{-1}(\frac{3}{4}) = \theta\) and \( cos^{-1}(\frac{1}{4}) = x\)

First solving for first part:
\[\sin^{-1}(\frac{3}{4}) = \theta\]
\[\sin(\theta) = \frac{3}{4}\]
\[\cos(\theta) = \sqrt{1 - \sin^2(\theta)} \implies \cos(\theta) = \sqrt{1 - (\frac{3}{4})^2} \implies \cos(\theta) = \frac{\sqrt{7}}{4}\]

Answer 2

Similarly can you do it for second part:
\[\cos^{-1}(\frac{1}{4}) = x\]

can you find sin(x) here?

Answer 3

@sfb you there?