Question

use the given value of a trigonometric function of θ to find the values of the other trigonmetric functions. assume θ is an acute angle. answer exactly.

(sin(\theta)=\frac{3}{5})

(cos(\theta)=)

(\tan(\theta)=)

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Explanation:

Step1: Use Pythagorean identity

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

Step2: Use the definition of tangent

The definition of \(\tan\theta\) is \(\tan\theta=\frac{\sin\theta}{\cos\theta}\).
We know \(\sin\theta=\frac{3}{5}\) and \(\cos\theta=\frac{4}{5}\), then \(\tan\theta=\frac{\frac{3}{5}}{\frac{4}{5}}=\frac{3}{4}\).

Answer:

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