question 8
evaluate $ \cos \left( \sin ^ { - 1 } \left( \frac { 5 } { 9 } \
ight) \
ight) $, giving your answer as an exact value (no decimals).
question help: video

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Accepted Answer

# Explanation:
## Step1: Let $\theta=\sin^{-1}(\frac{5}{9})$
By the definition of the inverse - sine function, $\sin\theta=\frac{5}{9}$, and $\theta\in[-\frac{\pi}{2},\frac{\pi}{2}]$. In the interval $[-\frac{\pi}{2},\frac{\pi}{2}]$, $\cos\theta\geq0$.
## Step2: Use the Pythagorean identity $\sin^{2}\theta+\cos^{2}\theta = 1$
We know that $\cos\theta=\sqrt{1 - \sin^{2}\theta}$. Substitute $\sin\theta=\frac{5}{9}$ into the formula:
$$
\begin{align*}
\cos\theta&=\sqrt{1-(\frac{5}{9})^{2}}\
&=\sqrt{1-\frac{25}{81}}\
&=\sqrt{\frac{81 - 25}{81}}\
&=\sqrt{\frac{56}{81}}\
&=\frac{\sqrt{4\times14}}{9}\
&=\frac{2\sqrt{14}}{9}
\end{align*}
$$

# Answer:
$\frac{2\sqrt{14}}{9}$

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QUESTION IMAGE

Question

Explanation:

Step1: Let \(\theta=\sin^{-1}(\frac{5}{9})\)

Step2: Use the Pythagorean identity \(\sin^{2}\theta+\cos^{2}\theta = 1\)

Answer: