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 \\(\square\\). (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Apply Pythagorean Theorem

For right triangle $ABC$, $c^2 = a^2 + b^2$. Substitute $a=15$, $b=20$:
$$c^2 = 15^2 + 20^2$$

Step2: Calculate squared side lengths

Compute the squares:
$$c^2 = 225 + 400$$

Step3: Sum and solve for $c$

Add values, take positive root:
$$c^2 = 625 \implies c = \sqrt{625} = 25$$

Step4: Define trigonometric ratios

For $\theta$ at $A$:

• Opposite side $a=15$, adjacent side $b=20$, hypotenuse $c=25$

Step5: Compute sine of $\theta$

$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$:
$$\sin\theta = \frac{15}{25} = \frac{3}{5}$$

Step6: Compute cosine of $\theta$

$\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$:
$$\cos\theta = \frac{20}{25} = \frac{4}{5}$$

Step7: Compute tangent of $\theta$

$\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$:
$$\tan\theta = \frac{15}{20} = \frac{3}{4}$$

Step8: Compute cosecant of $\theta$

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

Step9: Compute secant of $\theta$

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

Step10: Compute cotangent of $\theta$

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

Answer:

Missing side length: $25$
$\sin\theta = \frac{3}{5}$, $\cos\theta = \frac{4}{5}$, $\tan\theta = \frac{3}{4}$, $\csc\theta = \frac{5}{3}$, $\sec\theta = \frac{5}{4}$, $\cot\theta = \frac{4}{3}$