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Exercises Trigonometry

Example 4 from Algebra and Trigonometry, 9.2 Sum and Difference Identities

Finding the Exact Value of an Expression Involving an Inverse Trigonometric Function

Find the exact value of $\sin (\cos^{−1} \frac{1}{2} + \sin^{−1} \frac{3}{5}) .$ Then check the answer with a graphing calculator.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
The pattern displayed in this problem is $\sin (α + β) .$ Let $α = \cos^{−1} \frac{1}{2}$ and $β = \sin^{−1} \frac{3}{5} .$ Then we can write
$\begin{matrix} \cos α & = & \frac{1}{2}, 0 \leq α \leq π \\ \sin β & = & \frac{3}{5}, - \frac{π}{2} \leq β \leq \frac{π}{2} \end{matrix}$
We will use the Pythagorean identities to find $\sin α$ and $\cos β .$

$\begin{matrix} \sin α & = & \sqrt{1 - \cos^{2} α} \\ = & \sqrt{1 - \frac{1}{4}} \\ = & \sqrt{\frac{3}{4}} \\ = & \frac{\sqrt{3}}{2} \\ \cos β & = & \sqrt{1 - \sin^{2} β} \\ = & \sqrt{1 - \frac{9}{25}} \\ = & \sqrt{\frac{16}{25}} \\ = & \frac{4}{5} \end{matrix}$
Using the sum formula for sine,

$\begin{matrix} \sin (\cos^{- 1} \frac{1}{2} + \sin^{- 1} \frac{3}{5}) & = & \sin (α + β) \\ = & \sin α \cos β + \cos α \sin β \\ = & \frac{\sqrt{3}}{2} \cdot \frac{4}{5} + \frac{1}{2} \cdot \frac{3}{5} \\ = & \frac{4 \sqrt{3} + 3}{10} \end{matrix}$

How did it go?

PreviousUsing Sum and Difference Identities to Evaluate the Difference of Angles Next Finding the Exact Value of an Expression Involving Tangent

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 9.2, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.