Question 43·Medium·Geometry and Trigonometry

Angle $B$ is at vertex $B$. The side opposite an angle is the side that does not touch that angle. Is that $AC$ or $BC$?

For a right triangle, $\sin(\text{angle})$ is the ratio of the side opposite the angle to the hypotenuse. Write $\sin B$ as a fraction using the side lengths you know, then simplify the fraction.

In the Desmos expression line, type 6/10 to represent $\dfrac{\text{opposite}}{\text{hypotenuse}}$ for angle $B$. Note the decimal value Desmos gives.

In separate lines, type each fraction from the answer choices (for example, one line with 2/5, another with 3/5, and so on) and check their decimal values. The correct choice is the one whose decimal value matches the decimal from 6/10.

In a right triangle, the hypotenuse is the side opposite the right angle. Since angle $C$ is $90^\circ$, side $AB$ is the hypotenuse. For angle $B$, the side directly across from it (not touching $B$) is $AC$, so $AC$ is the side opposite angle $B$.

In a right triangle, $\sin(\text{angle}) = \dfrac{\text{opposite}}{\text{hypotenuse}}$. For angle $B$ in triangle $ABC$, this means

We are given $AC = 6$ and $AB = 10$, so

Simplify $6/10$ by dividing the numerator and denominator by $2$:

So $\sin B = \dfrac{3}{5}$, which corresponds to choice B.