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 using Pythagorean theorem

Let the adjacent side be $x$. By the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $c = 8$ (hypotenuse) and $b=7$ (opposite side to $\theta$). So $x=\sqrt{8^{2}-7^{2}}=\sqrt{64 - 49}=\sqrt{15}$.

Step2: Calculate $\sin\theta$

The definition of $\sin\theta$ in a right - triangle is $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. Here, $\sin\theta=\frac{7}{8}$.

Step3: Calculate $\csc\theta$

Since $\csc\theta=\frac{1}{\sin\theta}$, then $\csc\theta=\frac{8}{7}$.

Step4: Calculate $\cot\theta$

The definition of $\cot\theta$ in a right - triangle is $\cot\theta=\frac{\text{adjacent}}{\text{opposite}}$. So $\cot\theta=\frac{\sqrt{15}}{7}$.

Answer:

$\sin\theta=\frac{7}{8}$, $\csc\theta=\frac{8}{7}$, $\cot\theta=\frac{\sqrt{15}}{7}$