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Question

Difficulty: MediumLaw of Sines and Law of Cosines

In △ABC\triangle ABC, the length of side aa (opposite angle AA) is 99 inches, the length of side bb (opposite angle BB) is 1212 inches, and the measure of angle BB is 45∘45^\circ. What is the exact value of sin⁡A\sin A?

  1. 328\frac{3\sqrt{2}}{8}Answer
  2. B
    223\frac{2\sqrt{2}}{3}
  3. C
    324\frac{3\sqrt{2}}{4}
  4. D
    38\frac{3}{8}
  5. E
    423\frac{4\sqrt{2}}{3}

Answer

The exact value of sin⁡A\sin A is 328\frac{3\sqrt{2}}{8}.
According to the Law of Sines, sin⁡Aa=sin⁡Bb\frac{\sin A}{a} = \frac{\sin B}{b}. Substituting a=9a = 9, b=12b = 12, and B=45∘B = 45^\circ yields sin⁡A9=sin⁡45∘12\frac{\sin A}{9} = \frac{\sin 45^\circ}{12}. Multiplying both sides by 99 and substituting sin⁡45∘=22\sin 45^\circ = \frac{\sqrt{2}}{2} gives sin⁡A=9⋅2212=9224=328\sin A = \frac{9 \cdot \frac{\sqrt{2}}{2}}{12} = \frac{9\sqrt{2}}{24} = \frac{3\sqrt{2}}{8}.

Step-by-Step Solution

1
State the Law of Sines for the given triangle components.
sin⁡Aa=sin⁡Bb\frac{\sin A}{a} = \frac{\sin B}{b}
The Law of Sines relates the sines of angles to their opposite side lengths in any triangle.
2
Substitute the known values (a=9a = 9, b=12b = 12, and B=45∘B = 45^\circ) into the formula.
sin⁡A9=sin⁡45∘12\frac{\sin A}{9} = \frac{\sin 45^\circ}{12}
Plugging in the given numbers isolates the unknown quantity sin⁡A\sin A.
3
Solve for sin⁡A\sin A and substitute the exact value of sin⁡45∘=22\sin 45^\circ = \frac{\sqrt{2}}{2}.
sin⁡A=9⋅sin⁡45∘12=9⋅2212=9224=328\sin A = \frac{9 \cdot \sin 45^\circ}{12} = \frac{9 \cdot \frac{\sqrt{2}}{2}}{12} = \frac{9\sqrt{2}}{24} = \frac{3\sqrt{2}}{8}
Simplifying the fraction gives the exact trigonometric ratio.

Key Concept

Law of Sines
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Topics

Law of Sines and Law of Cosines

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