Problem 99 - Precalculus Essentials

Question

Find the exact value of each expression. Do not use a calculator. $$\sin \left(\cos ^{-1} \frac{1}{2}+\sin ^{-1} \frac{3}{5}\right)$$

Step-by-Step Solution

Answer

The exact value of the given expression is \( \frac{2\sqrt{3} + 3}{10} \).

Step 1: Identify the trigonometric identities

We need to find the exact value of \( \sin \left( \cos ^{-1} \frac{1}{2}+ \sin ^{-1} \frac{3}{5} \right) \). We can recognize that this can be evaluated using the sine addition formula \( \sin (A+B) =\sin A \cos B + \cos A \sin B \) where \( A=\cos ^{-1} \frac{1}{2} \) and \( B= \sin ^{-1} \frac{3}{5} \).

Step 2: Evaluate the sin and cos value of A and B

We know that, \( \sin \left( \cos ^{-1} x \right) =\sqrt{1-x^2} \) for \( -1 \leq x \leq 1 \) and \( \cos \left( \sin^{-1} x \right) = \sqrt{1-x^2} \) for \( -1 \leq x \leq 1 \). So, \( \sin A = \sin \left( \cos ^{-1} \frac{1}{2} \right) = \sqrt{1-\left(\frac{1}{2}\right)^2} = \frac{\sqrt{3}}{2} \), \( \cos A = \cos \left(\cos ^{-1} \frac{1}{2}\right)=\frac{1}{2} \), \( \sin B = \sin \left(\sin ^{-1} \frac{3}{5}\right)=\frac{3}{5} \) and \( \cos B = \cos \left(\sin ^{-1} \frac{3}{5}\right)=\sqrt{1-\left(\frac{3}{5}\right)^2}= \frac{4}{5} \).

Step 3: Substitute into the sine addition formula

Now replace \( A, B, \sin A, \cos A, \sin B, \cos B \) in addition formula, we get \( \sin (A+B) =\sin A \cos B + \cos A \sin B = \left(\frac{\sqrt{3}}{2}\right) \left(\frac{4}{5}\right) + \left(\frac{1}{2}\right) \left(\frac{3}{5}\right) \): this gives \( \sin (A+B) =\frac{2\sqrt{3}}{5} + \frac{3}{10} = \frac{2\sqrt{3} + 3}{10} \).