Question 29·Medium·Right Triangles and Trigonometry

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In right triangle $\triangle ABC$ , $\angle C$ is a right angle, $\angle A$ measures $35^{\circ}$ , and side $AC$ (the leg adjacent to $\angle A$ ) has length $12$ units.

Which expression represents the length of the hypotenuse $AB$ ?

For right-triangle trig questions, first label the sides relative to the given angle (opposite, adjacent, hypotenuse). Then pick the trig function that uses the side you know and the side you need (SOH-CAH-TOA), write the basic ratio equation, and solve algebraically for the unknown. As a quick reasonableness check, remember the hypotenuse must be the longest side, so if your expression clearly gives a value smaller than the known leg, you likely multiplied instead of dividing by a trig value less than 1.

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Label the sides in the triangle

Which side is the hypotenuse, and which side is adjacent to angle $A$ ? Use the information that angle $C$ is the right angle and $AC = 12$ .

Pick the right trig ratio

You know the adjacent side to angle $A$ and want the hypotenuse. From SOH-CAH-TOA, which trig function connects the adjacent side and the hypotenuse?

Set up and solve the equation

Write an equation using that trig function with angle $35^{\circ}$ , the known side 12, and the unknown hypotenuse. Then rearrange the equation to isolate the hypotenuse.

Desmos Guide

Check each option numerically

Make sure Desmos is in degree mode. In separate lines, enter the four expressions exactly as written: 12*cos(35), 12*tan(35), 12/tan(35), and 12/cos(35). Compare the numerical outputs and remember that the hypotenuse must be longer than 12, since 12 is a leg; focus on which expressions give values that make sense for the hypotenuse.

Step-by-step Explanation

Identify the sides relative to angle A

In right triangle $\triangle ABC$ , angle $C$ is the right angle, so side $AB$ is the hypotenuse (the side opposite the right angle).

Angle $A$ is $35^{\circ}$ and side $AC$ is given as 12 units adjacent to angle $A$ . So:

Adjacent side to angle $A$ : $AC = 12$
Hypotenuse: $AB$ (unknown)
Opposite side to angle $A$ : $BC$ (not used here)

Choose the correct trig ratio

For a right triangle and an acute angle, remember SOH-CAH-TOA:

$\sin(\theta) = \dfrac{\text{opposite}}{\text{hypotenuse}}$
$\cos(\theta) = \dfrac{\text{adjacent}}{\text{hypotenuse}}$
$\tan(\theta) = \dfrac{\text{opposite}}{\text{adjacent}}$

We know the adjacent side ( $AC$ ) and want the hypotenuse ( $AB$ ), so we use cosine:

\cos 35^{\circ} = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{AC}{AB} = \frac{12}{AB}.

Solve the cosine equation for the hypotenuse

We have

\cos 35^{\circ} = \frac{12}{AB}.

Solve for $AB$ :

Multiply both sides by $AB$ :

AB \cdot \cos 35^{\circ} = 12.

Divide both sides by $\cos 35^{\circ}$ :

AB = \frac{12}{\cos 35^{\circ}}.

So the expression that represents the length of the hypotenuse $AB$ is $\dfrac{12}{\cos 35^{\circ}}$ , which corresponds to choice D.

Question 29·Medium·Right Triangles and Trigonometry

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation