Question

use the given value of a trigonometric function of \\( \theta \\) to find the values of the other trigonometric functions. assume \\( \theta \\) is an acute angle. answer exactly.
\\( \sin (\theta)=\frac{12}{15} \\)
\\( \cos (\theta)= \\)
\\( \tan (\theta)= \\)

Explanation:

Step1: Use Pythagorean identity

We know that \(\sin^{2}\theta+\cos^{2}\theta = 1\). Given \(\sin\theta=\frac{12}{15}\), then \(\cos^{2}\theta=1-\sin^{2}\theta\).
Substitute \(\sin\theta\): \(\cos^{2}\theta = 1-(\frac{12}{15})^{2}=1-\frac{144}{225}=\frac{225 - 144}{225}=\frac{81}{225}\).
Since \(\theta\) is acute, \(\cos\theta>0\), so \(\cos\theta=\sqrt{\frac{81}{225}}=\frac{9}{15}=\frac{3}{5}\).

Step2: Use tangent formula

The formula for \(\tan\theta\) is \(\tan\theta=\frac{\sin\theta}{\cos\theta}\).
Substitute \(\sin\theta=\frac{12}{15}\) and \(\cos\theta=\frac{9}{15}\): \(\tan\theta=\frac{\frac{12}{15}}{\frac{9}{15}}=\frac{12}{9}=\frac{4}{3}\).

Answer:

\(\cos(\theta)=\frac{3}{5}\), \(\tan(\theta)=\frac{4}{3}\)