Question

1. if the sine of angle β of a right - triangle is $\frac{1}{3}$, and the side adjacent to that angle measures $5sqrt{2}$ units, which of the following must be true?

a) $cos\beta=\frac{5sqrt{2}}{3}$
b) $cos\beta=\frac{2sqrt{2}}{3}$
c) $\tan\beta=\frac{1}{5sqrt{2}}$
d) $\tan\beta=\frac{1}{5sqrt{2}}$

Explanation:

Step1: Recall sine - cosine - tangent relationships

We know that $\sin\beta=\frac{opposite}{hypotenuse}=\frac{1}{3}$, let the opposite side be $a = 1$ and the hypotenuse be $c = 3$. Using the Pythagorean theorem $c^{2}=a^{2}+b^{2}$, where $b$ is the adjacent side. Given $b = 5\sqrt{2}$. Also, $\cos\beta=\frac{adjacent}{hypotenuse}$ and $\tan\beta=\frac{opposite}{adjacent}$.

Step2: Calculate cosine of the angle

$\cos\beta=\frac{adjacent}{hypotenuse}$. Since the adjacent side to angle $\beta$ is $5\sqrt{2}$ and the hypotenuse $c = 3$ (from $\sin\beta=\frac{1}{3}$), $\cos\beta=\frac{5\sqrt{2}}{3}$.

Step3: Calculate tangent of the angle

$\tan\beta=\frac{opposite}{adjacent}=\frac{1}{5\sqrt{2}}=\frac{\sqrt{2}}{10}$ (rationalized form). But we are mainly interested in cosine here as per the options.

Answer:

A. $\cos\beta=\frac{5\sqrt{2}}{3}$