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Difficulty: EasyRight Triangle Trigonometry (SOHCAHTOA)

In right triangle ABCABC, the right angle is at vertex CC. The length of leg ACAC is 1212 inches. If sin⁡(B)=35\sin(B) = \frac{3}{5}, what is the length, in inches, of the hypotenuse ABAB?

  1. A
    9
  2. B
    15
  3. C
    16
  4. 20Answer
  5. E
    25

Answer

The length of the hypotenuse ABAB is 20 inches.
The sine of angle BB is defined as the ratio of the opposite side to the hypotenuse, which is sin⁡(B)=ACAB\sin(B) = \frac{AC}{AB}. Substituting the given values, we get 35=12AB\frac{3}{5} = \frac{12}{AB}. Solving for the hypotenuse ABAB gives 3⋅AB=603 \cdot AB = 60, which simplifies to AB=20AB = 20.

Step-by-Step Solution

1
Identify the trigonometric ratio for sine in a right triangle.
sin⁡(B)=oppositehypotenuse=ACAB\sin(B) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{AC}{AB}
By definition of right triangle trigonometry (SOHCAHTOA), the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.
2
Substitute the given values into the sine ratio equation.
35=12AB\frac{3}{5} = \frac{12}{AB}
The opposite side to angle BB is leg ACAC, which has a length of 1212 inches, and sin⁡(B)\sin(B) is given as 35\frac{3}{5}.
3
Solve the proportion for the hypotenuse ABAB.
3⋅AB=60  ⟹  AB=203 \cdot AB = 60 \implies AB = 20
Cross-multiplying yields 3⋅AB=5⋅12=603 \cdot AB = 5 \cdot 12 = 60. Dividing both sides by 33 gives the length of the hypotenuse.

Key Concept

Right Triangle Trigonometry (SOHCAHTOA)
Estimated Time:45s
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Topics

Right Triangle Trigonometry (SOHCAHTOA)

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