Question

based on the given figure, find the following values of trigonometric functions. your answers should be numbers. note that in the graph, \alpha\ stands for $alpha$, and \beta\ stands for $\beta$. as usual with webwork questions, you do not need a calculator (and should not use one)
$sinalpha=\frac{4}{9}$
$sin\beta=\frac{8}{9}$
$cosalpha=\frac{8}{9}$
$cos\beta=\frac{4}{9}$
$\tanalpha=\frac{1}{2}$
$\tan\beta = 2$
$secalpha=\frac{9}{4}$
$cscalpha=\frac{9}{8}$
$cotalpha = 2$
note: you can earn partial credit on this problem.

Explanation:

Step1: Recall sine - cosine - tangent definitions

For a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$, $\sec\theta=\frac{1}{\cos\theta}$, $\csc\theta=\frac{1}{\sin\theta}$, $\cot\theta=\frac{1}{\tan\theta}$.
For angle $\alpha$:
The opposite side to $\alpha$ is 4, the adjacent side is 8, and the hypotenuse is 9.
$\sin\alpha=\frac{4}{9}$ (given by $\frac{\text{opposite}}{\text{hypotenuse}}$).
$\cos\alpha=\frac{8}{9}$ (by $\frac{\text{adjacent}}{\text{hypotenuse}}$).
$\tan\alpha=\frac{\sin\alpha}{\cos\alpha}=\frac{4/9}{8/9}=\frac{4}{8}=\frac{1}{2}$.
$\sec\alpha=\frac{1}{\cos\alpha}=\frac{9}{8}$.
$\csc\alpha=\frac{1}{\sin\alpha}=\frac{9}{4}$.
$\cot\alpha=\frac{1}{\tan\alpha}=2$.
For angle $\beta$:
The opposite side to $\beta$ is 8, the adjacent side is 4, and the hypotenuse is 9.
$\sin\beta=\frac{8}{9}$ (by $\frac{\text{opposite}}{\text{hypotenuse}}$).
$\cos\beta=\frac{4}{9}$ (by $\frac{\text{adjacent}}{\text{hypotenuse}}$).
$\tan\beta=\frac{\sin\beta}{\cos\beta}=\frac{8/9}{4/9}=\frac{8}{4} = 2$.

Answer:

$\sin\alpha=\frac{4}{9}$, $\sin\beta=\frac{8}{9}$, $\cos\alpha=\frac{8}{9}$, $\cos\beta=\frac{4}{9}$, $\tan\alpha=\frac{1}{2}$, $\tan\beta = 2$, $\sec\alpha=\frac{9}{8}$, $\csc\alpha=\frac{9}{4}$, $\cot\alpha=2$