Question

find tanθ, sinθ, and secθ, where θ is the angle shown in the figure. give exact values, not decimal approximations.

Explanation:

Step1: Recall tangent formula

$\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. In the right - triangle, the opposite side to $\theta$ is 7 and the adjacent side is 4. So, $\tan\theta = \frac{7}{4}$.

Step2: Find the hypotenuse

Use the Pythagorean theorem $c^{2}=a^{2}+b^{2}$, where $a = 4$ and $b = 7$. Then $c=\sqrt{4^{2}+7^{2}}=\sqrt{16 + 49}=\sqrt{65}$.

Step3: Recall sine formula

$\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. So, $\sin\theta=\frac{7}{\sqrt{65}}=\frac{7\sqrt{65}}{65}$.

Step4: Recall secant formula

$\sec\theta=\frac{\text{hypotenuse}}{\text{adjacent}}$. So, $\sec\theta=\frac{\sqrt{65}}{4}$.

Answer:

$\tan\theta=\frac{7}{4}$, $\sin\theta=\frac{7\sqrt{65}}{65}$, $\sec\theta=\frac{\sqrt{65}}{4}$