Question

task #1 - primary trigonometric ratios

1. determine the sin θ, cos θ and tan θ. then determine the measure of angle θ.

Explanation:

Step1: Recall trig - ratio definitions

In a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$.
The side opposite to $\theta$ is 4, the side adjacent to $\theta$ is 7, and the hypotenuse is $\sqrt{65}$.

Step2: Calculate $\sin\theta$

$\sin\theta=\frac{4}{\sqrt{65}}=\frac{4\sqrt{65}}{65}$

Step3: Calculate $\cos\theta$

$\cos\theta=\frac{7}{\sqrt{65}}=\frac{7\sqrt{65}}{65}$

Step4: Calculate $\tan\theta$

$\tan\theta=\frac{4}{7}$

Step5: Find the measure of $\theta$

$\theta=\arctan(\frac{4}{7})\approx29.74^{\circ}$

Answer:

$\sin\theta=\frac{4\sqrt{65}}{65}$, $\cos\theta=\frac{7\sqrt{65}}{65}$, $\tan\theta = \frac{4}{7}$, $\theta\approx29.74^{\circ}$