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Difficulty: MediumRight Triangle Trigonometry (SOHCAHTOA)

A surveyor standing at point PP on level ground measures the angle of elevation to the top of a cliff, point TT, as θ\theta. The ground distance from point PP to the vertical base of the cliff, point BB, is 120120 feet. If cos⁡(θ)=1213\cos(\theta) = \frac{12}{13}, what is the vertical height of the cliff, in feet?

  1. 50Answer
  2. B
    65
  3. C
    110
  4. D
    130
  5. E
    288

Answer

The vertical height of the cliff is 50 feet.
In right triangle △PBT\triangle PBT with the right angle at BB, point PP on the ground forms angle θ\theta. The adjacent side PB=120PB = 120 feet. Given cos⁡(θ)=adjacenthypotenuse=1213\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{12}{13}, we solve for hypotenuse PT=130PT = 130 feet. Using the Pythagorean theorem (1302−1202=502130^2 - 120^2 = 50^2), the opposite side TBTB (the vertical height of the cliff) is 5050 feet.

Step-by-Step Solution

1
Identify the given trigonometric ratio and sides of the right triangle △PBT\triangle PBT.
Angle θ\theta is at vertex PP, the adjacent leg PB=120PB = 120 feet, and cos⁡(θ)=adjacenthypotenuse=1213\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{12}{13}.
Cosine relates the adjacent side to the hypotenuse in a right triangle.
2
Calculate the length of the hypotenuse PTPT.
120PT=1213  ⟹  PT=120×1312=130\frac{120}{PT} = \frac{12}{13} \implies PT = \frac{120 \times 13}{12} = 130 feet.
Solve the ratio for the hypotenuse PTPT.
3
Find the vertical height TBTB using the Pythagorean theorem or sine ratio.
TB=PT2−PB2=1302−1202=16900−14400=2500=50TB = \sqrt{PT^2 - PB^2} = \sqrt{130^2 - 120^2} = \sqrt{16900 - 14400} = \sqrt{2500} = 50 feet.
In right triangle △PBT\triangle PBT, TB2+PB2=PT2TB^2 + PB^2 = PT^2.

Key Concept

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

Right Triangle Trigonometry (SOHCAHTOA)

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