Question

use the given value of a trigonometric function of \\( \theta \\) to find the values of the other trigonmetric functions. assume \\( \theta \\) is an acute angle. answer exactly.
\\( \tan (\theta)=3 \\)
\\( \sin (\theta)= \\)
\\( \cos (\theta)= \\)
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Explanation:

Step1: Recall the Pythagorean identity

We know that \(\tan\theta=\frac{\sin\theta}{\cos\theta} = 3\), so \(\sin\theta=3\cos\theta\). Also, from the Pythagorean identity \(\sin^{2}\theta+\cos^{2}\theta = 1\).

Step2: Substitute \(\sin\theta = 3\cos\theta\) into the Pythagorean identity

Substitute into \(\sin^{2}\theta+\cos^{2}\theta=1\), we get \((3\cos\theta)^{2}+\cos^{2}\theta=1\).

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Step3: Find \(\sin\theta\)

Since \(\sin\theta = 3\cos\theta\), and \(\cos\theta=\frac{\sqrt{10}}{10}\), then \(\sin\theta=3\times\frac{\sqrt{10}}{10}=\frac{3\sqrt{10}}{10}\)

Answer:

\(\sin(\theta)=\frac{3\sqrt{10}}{10}\), \(\cos(\theta)=\frac{\sqrt{10}}{10}\)