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Question

Difficulty: MediumRight Triangle Trigonometry (SOHCAHTOA)

A straight ramp is constructed for a skateboard park. The ramp rises to a vertical height of 535\sqrt{3} feet above flat horizontal ground, and the horizontal distance from the base of the ramp to the point directly beneath its highest point is 1515 feet. If θ\theta represents the angle of inclination of the ramp with respect to the ground, what is the value of cos⁡(θ)\cos(\theta)?

  1. A
    12\frac{1}{2}
  2. B
    33\frac{\sqrt{3}}{3}
  3. 32\frac{\sqrt{3}}{2}Answer
  4. D
    233\frac{2\sqrt{3}}{3}
  5. E
    3\sqrt{3}

Answer

The value of cos⁡(θ)\cos(\theta) is 32\frac{\sqrt{3}}{2}.
The horizontal distance of 1515 feet is adjacent to angle θ\theta, and the vertical height of 535\sqrt{3} feet is opposite to θ\theta. By the Pythagorean theorem, the hypotenuse is 152+(53)2=225+75=103\sqrt{15^2 + (5\sqrt{3})^2} = \sqrt{225 + 75} = 10\sqrt{3} feet. Using SOHCAHTOA, cos⁡(θ)=adjacenthypotenuse=15103=32\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{15}{10\sqrt{3}} = \frac{\sqrt{3}}{2}.

Step-by-Step Solution

1
Identify the given side lengths relative to the angle θ\theta
The side opposite to angle θ\theta is 535\sqrt{3} feet, and the side adjacent to angle θ\theta is 1515 feet.
The height of the ramp is opposite to the angle of inclination, and the horizontal ground distance is adjacent.
2
Calculate the length of the hypotenuse using the Pythagorean theorem
Hypotenuse =152+(53)2=225+75=300=103= \sqrt{15^2 + (5\sqrt{3})^2} = \sqrt{225 + 75} = \sqrt{300} = 10\sqrt{3} feet.
Cosine is defined as adjacent divided by hypotenuse, so the hypotenuse length must be calculated first.
3
Apply the cosine ratio definition cos⁡(θ)=adjacenthypotenuse\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} and simplify
cos⁡(θ)=15103=323=336=32\cos(\theta) = \frac{15}{10\sqrt{3}} = \frac{3}{2\sqrt{3}} = \frac{3\sqrt{3}}{6} = \frac{\sqrt{3}}{2}.
Rationalizing the denominator yields the simplified exact ratio.

Key Concept

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

Right Triangle Trigonometry (SOHCAHTOA)

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