Use the definition $\sin A = \dfrac{\text{opposite}}{\text{hypotenuse}}$. Which sides of the triangle are opposite and hypotenuse for angle $A$, and what convenient values can you assign to them?

Once you assign values to the opposite side and hypotenuse using $\sin A = \dfrac{5}{13}$, use the Pythagorean theorem to find the length of the remaining leg.

Now, for angle $B$, which sides are opposite and adjacent? Use those side lengths to form $\tan B = \dfrac{\text{opposite}}{\text{adjacent}}$.

In the Desmos scientific calculator, make sure you are in degree mode (tap the DEG/RAD toggle if needed). Then type asin(5/13) to find the measure of angle $A$.

On a new line, type 90 - asin(5/13) to compute angle $B$ (since $A + B = 90$ in a right triangle). Desmos will display the approximate value of angle $B$ in degrees.

On another line, type tan(90 - asin(5/13)). Note the decimal value that Desmos gives for this expression; that is the value of $\tan B$.

Type each answer option into Desmos as a separate expression: 5/12, 12/5, 13/5, and 12/13. Compare their decimal values to the value you found for tan(90 - asin(5/13)) and select the choice whose decimal matches.

In a right triangle with right angle at $C$, the other two angles satisfy $A + B = 90^\circ$, so $A$ and $B$ are complementary.

We are given $\sin A = \dfrac{5}{13}$ and asked to find $\tan B$.

Recall the definitions for angle $A$:

• $\sin A = \dfrac{\text{opposite}}{\text{hypotenuse}}$
• $\tan A = \dfrac{\text{opposite}}{\text{adjacent}}$.

Let’s first figure out all three side lengths (up to a common scale factor) using $\sin A = \dfrac{5}{13}$. Then we can use those sides to get $\tan B$.

Angle $A$ is at vertex $A$.

• The side opposite angle $A$ is $BC$.
• The hypotenuse of the right triangle is $AB$.

Since $\sin A = \dfrac{\text{opposite}}{\text{hypotenuse}} = \dfrac{5}{13}$, we can set:

• $BC = 5k$
• $AB = 13k$ for some positive scale factor $k$.

Now use the Pythagorean theorem to find the third side $AC$:

So the three sides are $5k$, $12k$, and $13k$ (a 5-12-13 right triangle).

Now focus on angle $B$ (at vertex $B$).

• The side opposite angle $B$ is $AC$.
• The side adjacent to angle $B$ (but not the hypotenuse) is $BC$.

By definition:

From Step 2, we know:

• $AC = 12k$
• $BC = 5k$

So at this point, $\tan B$ can be written using these expressions for the sides.

Substitute the values of $AC$ and $BC$ into the expression for $\tan B$:

The common factor $k$ cancels:

So the correct answer is $\dfrac{12}{5}$, which corresponds to choice B.