Question

find \\( \cos \theta \\), \\( \tan \theta \\), and \\( \csc \theta \\), where \\( \theta \\) is the angle shown in the figure. give exact values, not decimal approximations. \\( \cos \theta = \\) \\( \tan \theta = \\) \\( \csc \theta = \\)

Explanation:

Step1: Find the hypotenuse

Using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = 4\), \(b = 5\). Then \(c=\sqrt{4^{2}+5^{2}}=\sqrt{16 + 25}=\sqrt{41}\).

Step2: Calculate \(\cos\theta\)

By the definition \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), so \(\cos\theta=\frac{4}{\sqrt{41}}=\frac{4\sqrt{41}}{41}\).

Step3: Calculate \(\tan\theta\)

By the definition \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), so \(\tan\theta=\frac{5}{4}\).

Step4: Calculate \(\csc\theta\)

By the definition \(\csc\theta=\frac{\text{hypotenuse}}{\text{opposite}}\), so \(\csc\theta=\frac{\sqrt{41}}{5}\).

Answer:

\(\cos\theta=\frac{4\sqrt{41}}{41}\), \(\tan\theta=\frac{5}{4}\), \(\csc\theta=\frac{\sqrt{41}}{5}\)