in exercises 43 - 48, evaluate without using a calculator.\n43. find \\( \\sin \\theta \\) and \\( \\tan…

Explanation:

Step1: Determine the quadrant

Since (\cos\theta=\frac{2}{3}>0) and (\cot\theta=\frac{\cos\theta}{\sin\theta}>0), then (\sin\theta>0). So (\theta) is in the first quadrant.

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

Substitute (\cos\theta=\frac{2}{3}) into the identity: (\sin^{2}\theta+\left(\frac{2}{3}\right)^{2}=1) (\sin^{2}\theta=1 - \frac{4}{9}=\frac{9 - 4}{9}=\frac{5}{9}) Since (\sin\theta>0), then (\sin\theta=\sqrt{\frac{5}{9}}=\frac{\sqrt{5}}{3})

Step3: Calculate (\tan\theta)

Use the formula (\tan\theta=\frac{\sin\theta}{\cos\theta}) Substitute (\sin\theta=\frac{\sqrt{5}}{3}) and (\cos\theta=\frac{2}{3}) (\tan\theta=\frac{\frac{\sqrt{5}}{3}}{\frac{2}{3}}=\frac{\sqrt{5}}{2})

Answer:

(\sin\theta=\frac{\sqrt{5}}{3}), (\tan\theta=\frac{\sqrt{5}}{2})