Chapter 4.7, Problem 75E

To determine

To find: The exact value of the expression $\sin \left(\arccos \left(-\frac{2}{3}\right)\right)$ . Use a graphing utility to verify the result.

Answer to Problem 75E

  $\sin \left(\arccos \left(-\frac{2}{3}\right)\right)=\frac{\sqrt{5}}{3}$

Explanation of Solution

Given:

  $\sin \left(\arccos \left(-\frac{2}{3}\right)\right)$

Calculation:

The given expression is $\sin \left(\arccos \left(-\frac{2}{3}\right)\right)$

This expression can be rewritten as $\sin \left(\arccos \left(\frac{2}{3}\right)\right)$

Let $\arccos \left(\frac{2}{3}\right)=y$ , then the given expression becomes $\sin y$

Since, $\arccos \left(\frac{2}{3}\right)=y\Rightarrow \cos y=\frac{2}{3}$

Using this information, make a right triangle as shown below

  

Using Pythagoras Theorem,

  $\begin{array}{c}AB=\sqrt{3^2-2^2}\\ =\sqrt{9-4}\\ =\sqrt{5}\end{array}$

From the right triangle,

  $\begin{array}{c}\sin y=\frac{AB}{AC}\\ =\frac{\sqrt{5}}{3}\end{array}$

Therefore,

  $\sin \left(\arccos \left(-\frac{2}{3}\right)\right)=\frac{\sqrt{5}}{3}$

Now, using the graphing utility, the value of the expression is $\frac{\sqrt{5}}{3}$

Since, the calculated value is equal to the value that is found using a graphing utility.

Hence, the calculated value is true and this verified the result.