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

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})