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Difficulty: EasyRight Triangle Trigonometry (SOHCAHTOA)

A straight skateboard ramp has a vertical height of 99 feet and a horizontal length of 1212 feet along the ground. The vertical height and horizontal length meet at a right angle. What is the sine of the angle that the ramp makes with the ground?

  1. A
    34\frac{3}{4}
  2. B
    45\frac{4}{5}
  3. 35\frac{3}{5}Answer
  4. D
    43\frac{4}{3}
  5. E
    53\frac{5}{3}

Answer

The value of the sine of the angle is 35\frac{3}{5}.
The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. The opposite side is the vertical height of 99 feet. Using the Pythagorean theorem, the hypotenuse (the length of the ramp) is calculated as 92+122=15\sqrt{9^2 + 12^2} = 15 feet. Therefore, the sine of the angle is 915\frac{9}{15}, which simplifies to 35\frac{3}{5}.

Step-by-Step Solution

1
Identify the lengths of the legs of the right triangle and calculate the length of the hypotenuse using the Pythagorean theorem.
The hypotenuse (ramp length) is 1515 feet.
The vertical height (99 feet) and horizontal length (1212 feet) form the perpendicular legs of a right triangle. The length of the ramp is the hypotenuse: 92+122=81+144=225=15\sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15 feet.
2
Identify the opposite side and hypotenuse relative to the angle the ramp makes with the ground, then apply the sine ratio definition.
The sine of the angle is 35\frac{3}{5}.
The angle the ramp makes with the ground is at the bottom vertex. The side opposite this angle is the vertical height (99 feet), and the hypotenuse is the ramp length (1515 feet). The sine ratio is sin⁡(θ)=oppositehypotenuse=915\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{9}{15}, which simplifies to 35\frac{3}{5}.

Key Concept

Right Triangle Trigonometry (SOHCAHTOA)
Estimated Time:1m 0s
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Topics

Right Triangle Trigonometry (SOHCAHTOA)

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