Question

1. the angle measures and side lengths of a triangle are shown here.

select all true statements:
a. \\(\sin(\theta) = \frac{4}{\sqrt{97}}\\) b. \\(\tan(\beta) = \frac{9}{4}\\) c. \\(\tan(\beta) = \frac{4}{9}\\) d. \\(\cos(\beta) = \frac{4}{\sqrt{97}}\\) e. \\(4^2 + 9^2 = 97\\)

Explanation:

Step1: Confirm triangle properties

This is a right triangle with $AC=9$, $BC=4$, $AB=\sqrt{97}$. Verify hypotenuse: $9^2+4^2=81+16=97$, so $AB=\sqrt{97}$ is correct.

Step2: Calculate $\sin(\theta)$

$\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{4}{\sqrt{97}}$ → matches option A.

Step3: Calculate $\tan(\beta)$

$\tan(\beta)=\frac{\text{opposite}}{\text{adjacent}}=\frac{9}{4}$ → matches option B, contradicts C.

Step4: Calculate $\cos(\beta)$

$\cos(\beta)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{4}{\sqrt{97}}$ → matches option D.

Step5: Verify option E

$4^2+9^2=16+81=97$ → matches option E.

Answer:

A. $\sin(\theta) = \frac{4}{\sqrt{97}}$, B. $\tan(\beta) = \frac{9}{4}$, D. $\cos(\beta) = \frac{4}{\sqrt{97}}$, E. $4^2 + 9^2 = 97$