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Difficulty: MediumRight Triangle Trigonometry (SOHCAHTOA)

A technician is installing a straight support beam for a solar panel array mounted on a flat roof. The beam forms a right triangle with the horizontal roof and a vertical panel frame. The vertical frame is 2424 inches tall, and the angle θ\theta between the support beam and the horizontal roof satisfies sin⁡(θ)=1213\sin(\theta) = \frac{12}{13}. What is the horizontal distance, in inches, along the roof from the bottom of the vertical frame to the anchor point of the support beam?

  1. A
    5
  2. 10Answer
  3. C
    12
  4. D
    24
  5. E
    26

Answer

The horizontal distance along the roof from the bottom of the frame to the anchor point is 10 inches.
In a right-angled triangle formed by the vertical frame, horizontal roof, and diagonal support beam, the angle θ\theta is between the beam and the roof. The vertical frame (24 inches) is the side opposite to θ\theta. Using the sine definition, sin⁡(θ)=oppositehypotenuse=1213\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{12}{13}, so 24hypotenuse=1213\frac{24}{\text{hypotenuse}} = \frac{12}{13}, which yields a hypotenuse length of 26 inches. Applying the Pythagorean theorem to find the horizontal adjacent leg gives 262−242=100=10\sqrt{26^2 - 24^2} = \sqrt{100} = 10 inches.

Step-by-Step Solution

1
Identify the sides of the right triangle relative to the angle θ\theta.
The vertical frame of length 24 inches is opposite to θ\theta, the support beam is the hypotenuse, and the horizontal distance along the roof is adjacent to θ\theta.
SOHCAHTOA defines sin⁡(θ)=oppositehypotenuse\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}.
2
Calculate the length of the hypotenuse using the sine ratio.
24hypotenuse=1213  ⟹  hypotenuse=24×1312=26\frac{24}{\text{hypotenuse}} = \frac{12}{13} \implies \text{hypotenuse} = 24 \times \frac{13}{12} = 26 inches.
Setting the opposite side (24) over hypotenuse equal to 1213\frac{12}{13} allows solving for the hypotenuse.
3
Calculate the horizontal adjacent side using the Pythagorean theorem.
adjacent=262−242=676−576=100=10\text{adjacent} = \sqrt{26^2 - 24^2} = \sqrt{676 - 576} = \sqrt{100} = 10 inches.
In a right triangle, a2+b2=c2a^2 + b^2 = c^2, so the unknown leg is c2−b2\sqrt{c^2 - b^2}.

Key Concept

Right Triangle Trigonometry (SOHCAHTOA)
Estimated Time:1m 15s
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Topics

Right Triangle Trigonometry (SOHCAHTOA)

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