Question

find \\( \sec \theta \\), \\( \sin \theta \\), and \\( \tan \theta \\), where \\( \theta \\) is the angle shown in the figure. give exact values, not decimal approximations.

the right triangle has hypotenuse 10, opposite side to \\( \theta \\) is 7, and right angle at the bottom right.

\\( \sec \theta = \square \\)
\\( \sin \theta = \square \\)
\\( \tan \theta = \square \\)

Explanation:

Step1: Find adjacent side

Let adjacent side = $x$. By Pythagoras: $x^2 + 7^2 = 10^2$ → $x^2 = 100 - 49 = 51$ → $x = \sqrt{51}$

Step2: Calculate $\sec\theta$

$\sec\theta = \frac{\text{hypotenuse}}{\text{adjacent}} = \frac{10}{\sqrt{51}}$

Step3: Calculate $\sin\theta$

$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{7}{10}$

Step4: Calculate $\tan\theta$

$\tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{7}{\sqrt{51}}$

Answer:

$\sec\theta = \frac{10}{\sqrt{51}}$, $\sin\theta = \frac{7}{10}$, $\tan\theta = \frac{7}{\sqrt{51}}$