Question

watch the video and then solve the problem given below. click here to watch the video. use the pythagorean theorem to find the length of the missing side of the right triangle. then find the value of each of the six trigonometric functions of θ.

Explanation:

Step1: Find the missing side length

By the Pythagorean Theorem $a^{2}+b^{2}=c^{2}$, where $c = 10$ and $b = 8$. We solve for $a$:
\[

$$\begin{align*} a&=\sqrt{c^{2}-b^{2}}\\ &=\sqrt{10^{2}-8^{2}}\\ &=\sqrt{100 - 64}\\ &=\sqrt{36}\\ &=6 \end{align*}$$

\]

Step2: Calculate sine function

$\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{a}{c}=\frac{6}{10}=\frac{3}{5}$

Step3: Calculate cosine function

$\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{b}{c}=\frac{8}{10}=\frac{4}{5}$

Step4: Calculate tangent function

$\tan\theta=\frac{\text{opposite}}{\text{adjacent}}=\frac{a}{b}=\frac{6}{8}=\frac{3}{4}$

Step5: Calculate cosecant function

$\csc\theta=\frac{1}{\sin\theta}=\frac{5}{3}$

Step6: Calculate secant function

$\sec\theta=\frac{1}{\cos\theta}=\frac{5}{4}$

Step7: Calculate cotangent function

$\cot\theta=\frac{1}{\tan\theta}=\frac{4}{3}$

Answer:

The missing side length $a = 6$.
$\sin\theta=\frac{3}{5}$, $\cos\theta=\frac{4}{5}$, $\tan\theta=\frac{3}{4}$, $\csc\theta=\frac{5}{3}$, $\sec\theta=\frac{5}{4}$, $\cot\theta=\frac{4}{3}$