QUESTION 31 Without using a calculator, find the exact value of $$ \sin heta $$ if $$ \cos heta=\frac{2}{3}, \quad 0 \leq heta

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QUESTION 31 Without using a calculator, find the exact value of $$ \sin 	heta $$ if $$ \cos 	heta=\frac{2}{3}, \quad 0 \leq 	heta<\frac{\pi}{2} $$

UpStudy AI · Accepted Answer

To find the exact value of $$ \sin \theta $$ given that $$ \cos \theta = \frac{2}{3} $$ and $$ 0 \leq \theta < \frac{\pi}{2} $$, follow these steps:

1. **Use the Pythagorean Identity:**
   
   The fundamental identity for sine and cosine is:
   $$
   \sin^2 \theta + \cos^2 \theta = 1
   $$
   
2. **Plug in the Known Value of $$ \cos \theta $$:**
   $$
   \sin^2 \theta + \left(\frac{2}{3}\right)^2 = 1
   $$
   $$
   \sin^2 \theta + \frac{4}{9} = 1
   $$
   
3. **Solve for $$ \sin^2 \theta $$:**
   $$
   \sin^2 \theta = 1 - \frac{4}{9}
   $$
   $$
   \sin^2 \theta = \frac{9}{9} - \frac{4}{9} = \frac{5}{9}
   $$
   
4. **Take the Square Root to Find $$ \sin \theta $$:**
   
   Since $$ 0 \leq \theta < \frac{\pi}{2} $$, $$ \sin \theta $$ is positive.
   $$
   \sin \theta = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3}
   $$

**Final Answer:**
$$
\sin \theta = \frac{\sqrt{5}}{3}
$$
$$ \sin \theta = \frac{\sqrt{5}}{3} $$

QUESTION 31 Without using a calculator, find the exact value of \( \sin \theta \) if \( \cos \theta=\frac{2}{3}, \quad 0 \leq \theta<\frac{\pi}{2} \)

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