Once you decide which legs are $3$ and $5$, use the Pythagorean theorem to find the hypotenuse.

Since angle $C$ is $90^\circ$, what is $A + B$? How are the sine of one angle and the cosine of its complement related?

Use your side lengths to find $\sin A$, then use the fact that $\cos B$ is equal to a trig function of $A$. Which ratio of the sides gives you that value?

In a Desmos expression line, enter h = sqrt(3^2 + 5^2). This uses the legs 3 and 5 implied by $\tan A = 3/5$ and gives you the hypotenuse $h$.

For angle $B$, the leg of length 3 is adjacent and $h$ is the hypotenuse, so in a new line type cosB = 3 / h. The decimal shown for cosB is the value of $\cos B$.

Type each answer choice’s expression (for example, 3/5, 5/sqrt(34), etc.) into separate lines in Desmos. Convert them to decimals and see which one matches the value of cosB; that matching choice is the correct answer.

In a right triangle, $\tan(\text{angle}) = \dfrac{\text{opposite}}{\text{adjacent}}$.

For angle $A$:

• Let the leg opposite angle $A$ be $3$.
• Let the leg adjacent to angle $A$ be $5$.

In triangle $ABC$ with right angle at $C$:

• Opposite $A$ is side $BC$, so take $BC = 3$.
• Adjacent to $A$ (and not the hypotenuse) is side $AC$, so take $AC = 5$.

Let the hypotenuse $AB = h$. The legs are $3$ and $5$, so:

Now the sides relative to angle $A$ are:

• Opposite: $3$
• Adjacent: $5$
• Hypotenuse: $\sqrt{34}$.

In any triangle, the sum of the angle measures is $180^\circ$. Here, angle $C$ is a right angle, so $C = 90^\circ$.

That means angles $A$ and $B$ must add to $90^\circ$:

So $A$ and $B$ are complementary angles. For complementary angles,

Therefore, once we know $\sin A$, we automatically know $\cos B$.

For angle $A$, $\sin(\text{angle}) = \dfrac{\text{opposite}}{\text{hypotenuse}}$.

So

From Step 3, $\cos B = \sin A$, so

Rationalize the denominator:

So the correct answer is $\dfrac{3\sqrt{34}}{34}$, which corresponds to choice C.