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.

\\( \sin ( \theta ) = \frac { 4 } { 5 } \\)

\\( \cos ( \theta ) = \\)

\\( \tan ( \theta ) = \\)

question help: \\( \square \\) message instructor

Explanation:

Step1: Use Pythagorean identity

We know that \(\sin^{2}\theta+\cos^{2}\theta = 1\). Given \(\sin\theta=\frac{4}{5}\), then \(\cos^{2}\theta=1-\sin^{2}\theta\). Substitute \(\sin\theta\) into the formula: \(\cos^{2}\theta=1 - (\frac{4}{5})^{2}=1-\frac{16}{25}=\frac{25 - 16}{25}=\frac{9}{25}\). Since \(\theta\) is acute, \(\cos\theta>0\), so \(\cos\theta=\frac{3}{5}\).

Step2: Use the definition of tangent

The definition of \(\tan\theta\) is \(\tan\theta=\frac{\sin\theta}{\cos\theta}\). Substitute \(\sin\theta = \frac{4}{5}\) and \(\cos\theta=\frac{3}{5}\) into the formula: \(\tan\theta=\frac{\frac{4}{5}}{\frac{3}{5}}=\frac{4}{3}\).

Answer:

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