Question 151·Medium·Geometry and Trigonometry

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In right triangle $ABC$ , angle $C$ is the right angle. If $\sin(A)=\dfrac{5}{13}$ , what is the value of $\tan(B)$ ?

For right-triangle trig questions where you’re given one trig ratio and asked for another involving the other acute angle, first remember that the two acute angles are complementary, so functions switch roles (e.g., tan of one is the reciprocal of tan of the other). Quickly build a right triangle using the given ratio as side lengths, use the Pythagorean theorem to find the missing leg, and then write the requested trig ratio using the correct opposite and adjacent sides. This triangle-based approach is usually faster and less error-prone than working with angle measures or memorizing many separate trig identities.

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Think about the relationship between angles A and B

In any right triangle, the two non-right angles add up to $90^\circ$ . How are angles $A$ and $B$ related if angle $C$ is the right angle?

Turn sin(A) into a picture of the triangle

Use $\sin(A) = \dfrac{5}{13}$ to label the side opposite $A$ and the hypotenuse with convenient numbers. Which side should be $5$ and which should be $13$ ?

Find the missing side

Once you know two sides of a right triangle (one of them the hypotenuse), use the Pythagorean theorem to find the third side. That third side will be both adjacent to $A$ and opposite $B$ .

Now focus on angle B

From angle $B$ 's perspective, identify which side is opposite and which is adjacent (not the hypotenuse), then form the ratio opposite/adjacent to get $\tan(B)$ .

Desmos Guide

Represent angle A with arcsin

In a new expression line, type: A = arcsin(5/13). Desmos will interpret this in radians and store that angle as A.

Use the fact that A and B are complementary

In the next line, type: B = pi/2 - A. This sets B to $90^\circ - A$ (since $\pi/2$ radians equals $90^\circ$ ).

Compute tan(B) numerically

In a new line, type: tan(B). Note the decimal value that Desmos gives for this expression.

Match the decimal to an answer choice

Type each answer choice fraction (5/12, 12/5, 5/13, 13/12) into separate lines in Desmos and compare their decimal values to the value of tan(B). The matching fraction is the correct answer choice.

Step-by-step Explanation

Relate angles A and B in a right triangle

In right triangle $ABC$ , angle $C$ is the right angle, so the other two angles are acute and must add to $90^\circ$ .

That means

A + B = 90^\circ,

so $B$ is the complement of $A$ and $B = 90^\circ - A$ .

Translate sin(A) into side lengths

By definition, for angle $A$ in a right triangle,

$\sin(A) = \dfrac{\text{opposite to } A}{\text{hypotenuse}}$ .

We are given $\sin(A) = \dfrac{5}{13}$ , so we can take:

side opposite angle $A$ to be $5$ ,
the hypotenuse to be $13$ .

Now the third side (adjacent to $A$ and opposite $B$ ) is still unknown.

Use the Pythagorean theorem to find the third side

Let $x$ be the length of the third side (adjacent to $A$ and opposite $B$ ). Since the triangle is right with hypotenuse $13$ :

\begin{aligned} 13^2 &= 5^2 + x^2 \\ 169 &= 25 + x^2 \\ x^2 &= 144 \\ x &= 12 \end{aligned}

So the sides of the triangle form a $5$ – $12$ – $13$ right triangle.

Write tan(B) using the correct sides

For angle $B$ , tangent is defined as

$\tan(B) = \dfrac{\text{opposite to } B}{\text{adjacent to } B}$ .

From angle $B$ 's point of view:

the side opposite $B$ is the side of length $12$ ,
the side adjacent to $B$ (but not the hypotenuse) is the side of length $5$ .

Therefore,

\tan(B) = \frac{12}{5},

which matches answer choice B.

Question 151·Medium·Geometry and Trigonometry

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Step-by-step Explanation

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Step-by-step Explanation