Question

find sinθ, cscθ, and cotθ, where θ is the angle shown in the figure. give exact values, not decimal approximations.

Explanation:

Step1: Find the adjacent - side length

Use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = 17\) (hypotenuse) and \(b = 15\) (opposite - side). Let the adjacent - side be \(a\). Then \(a=\sqrt{c^{2}-b^{2}}=\sqrt{17^{2}-15^{2}}=\sqrt{(17 + 15)(17 - 15)}=\sqrt{32\times2}=\sqrt{64}=8\).

Step2: Calculate \(\sin\theta\)

By the definition of sine in a right - triangle \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, the opposite side to \(\theta\) is \(15\) and the hypotenuse is \(17\), so \(\sin\theta=\frac{15}{17}\).

Step3: Calculate \(\csc\theta\)

Since \(\csc\theta=\frac{1}{\sin\theta}\), and \(\sin\theta=\frac{15}{17}\), then \(\csc\theta=\frac{17}{15}\).

Step4: Calculate \(\cot\theta\)

By the definition of cotangent in a right - triangle \(\cot\theta=\frac{\text{adjacent}}{\text{opposite}}\). Here, the adjacent side to \(\theta\) is \(8\) and the opposite side is \(15\), so \(\cot\theta=\frac{8}{15}\).

Answer:

\(\sin\theta=\frac{15}{17}\)
\(\csc\theta=\frac{17}{15}\)
\(\cot\theta=\frac{8}{15}\)