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

Want more problems? Check here!