find \\( \\sin \\theta \\) and \\( \\sec \\theta \\) if \\( \\tan \\theta = 3 \\).

Explanation:

Step1: Use the identity (1+\tan^{2}\theta=\sec^{2}\theta)

Given (\tan\theta = 3), substitute into the identity: [ \begin{align*} 1+(3)^{2}&=\sec^{2}\theta\ 1 + 9&=\sec^{2}\theta\ \sec^{2}\theta&=10\ \sec\theta&=\pm\sqrt{10} \end{align*} ]

Step2: Use the identity (\tan\theta=\frac{\sin\theta}{\cos\theta}) and (\cos\theta=\frac{1}{\sec\theta})

Since (\tan\theta = 3=\frac{\sin\theta}{\cos\theta}) and (\cos\theta=\frac{1}{\sec\theta}). If (\sec\theta=\sqrt{10}), then (\cos\theta=\frac{1}{\sqrt{10}}), and (\sin\theta=\tan\theta\times\cos\theta = 3\times\frac{1}{\sqrt{10}}=\frac{3}{\sqrt{10}}=\frac{3\sqrt{10}}{10}) If (\sec\theta=-\sqrt{10}), then (\cos\theta=-\frac{1}{\sqrt{10}}), and (\sin\theta=\tan\theta\times\cos\theta=3\times(-\frac{1}{\sqrt{10}})=-\frac{3}{\sqrt{10}}=-\frac{3\sqrt{10}}{10})

Answer:

If (\sec\theta=\sqrt{10}), then (\sin\theta=\frac{3\sqrt{10}}{10}); if (\sec\theta =-\sqrt{10}), then (\sin\theta=-\frac{3\sqrt{10}}{10})