Question

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. the length of the missing side of the right triangle is a = \square.

Explanation:

Step1: Recall Pythagorean Theorem

For right triangle: $a^2 + b^2 = c^2$

Step2: Rearrange to solve for $a$

$a = \sqrt{c^2 - b^2}$

Step3: Substitute given values

$a = \sqrt{34^2 - 30^2} = \sqrt{1156 - 900} = \sqrt{256}$

Step4: Calculate square root

$a = 16$

Step5: Define trigonometric ratios for $\theta$

• Opposite to $\theta$: $a=16$, Adjacent: $b=30$, Hypotenuse: $c=34$

Step6: Compute sine of $\theta$

$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{16}{34} = \frac{8}{17}$

Step7: Compute cosine of $\theta$

$\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{30}{34} = \frac{15}{17}$

Step8: Compute tangent of $\theta$

$\tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{16}{30} = \frac{8}{15}$

Step9: Compute cosecant of $\theta$

$\csc\theta = \frac{1}{\sin\theta} = \frac{17}{8}$

Step10: Compute secant of $\theta$

$\sec\theta = \frac{1}{\cos\theta} = \frac{17}{15}$

Step11: Compute cotangent of $\theta$

$\cot\theta = \frac{1}{\tan\theta} = \frac{15}{8}$

Answer:

Missing side length: $a = 16$
Trigonometric functions:
$\sin\theta = \frac{8}{17}$, $\cos\theta = \frac{15}{17}$, $\tan\theta = \frac{8}{15}$,
$\csc\theta = \frac{17}{8}$, $\sec\theta = \frac{17}{15}$, $\cot\theta = \frac{15}{8}$