For each angle θ listed below, find the reference angle α, and then find sin θ. Round sin θ to four decimal places, if necessary.
Ref. Angle a Sin θ
θ = 132°
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To find the reference angle α, we need to find the acute angle closest to θ that lies in the same quadrant.
In this case, since θ = 132°, it lies in the second quadrant. The acute angle closest to 132° in the second quadrant is 180° - 132° = 48°. Therefore, the reference angle α is 48°.
To find sin θ, we can refer to the unit circle. In the second quadrant, the sine value is positive.
sin θ = sin 132° = sin α = sin 48° ≈ 0.7431
To find the reference angle α for θ = 132°, we need to determine the acute angle that is formed between the terminal side of θ and the x-axis.
1. First, subtract 180° from θ to obtain an equivalent angle that lies in the interval between -180° and 180°:
θ - 180° = 132° - 180° = -48°
2. Since the resulting angle lies in the third quadrant, we need to find the angle in the first quadrant that has the same reference angle:
α = 180° - |-48°| = 180° - 48° = 132°
Now, we can find sin θ using the reference angle α:
3. sin θ = sin α
sin θ = sin 132°
To find sin 132°, you can use a calculator or reference tables. The value is approximately 0.9781 (rounded to four decimal places).