Find the Exact Value sin(157.5)

Problem

sin(157.5^∘)

Solution

1. Identify the angle as a half-angle of a known standard angle. Since 157.5^∘=315^∘/2 we can use the half-angle formula for sine.

2. Apply the formula for the half-angle of sine, which is sin(θ/2)=±√(,(1−cos(θ))/2)

3. Determine the sign based on the quadrant. The angle 157.5^∘ is in the second quadrant, where the sine function is positive.

4. Substitute θ=315^∘ into the formula.

sin(157.5^∘)=√(,(1−cos(315^∘))/2)

5. Evaluate the cosine of 315^∘ Since 315^∘ is in the fourth quadrant with a reference angle of 45^∘ cos(315^∘)=cos(45^∘)=√(,2)/2

6. Simplify the expression inside the square root.

sin(157.5^∘)=√(,(1−√(,2)/2)/2)

sin(157.5^∘)=√(,(2−√(,2))/2/2)

sin(157.5^∘)=√(,(2−√(,2))/4)

7. Extract the square root of the denominator.

sin(157.5^∘)=√(,2−√(,2))/2

Final Answer

sin(157.5^∘)=√(,2−√(,2))/2

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