Find the Exact Value cos(157.5)

Problem

cos(157.5^∘)

Solution

1. Identify the angle as a half-angle of a known standard angle.

157.5^∘=315^∘/2

2. Apply the half-angle formula for cosine, which is cos(θ/2)=±√(,(1+cos(θ))/2)

cos(157.5^∘)=±√(,(1+cos(315^∘))/2)

3. Determine the sign based on the quadrant. Since 157.5^∘ is in Quadrant II, the cosine value must be negative.

cos(157.5^∘)=−√(,(1+cos(315^∘))/2)

4. Evaluate the cosine of the reference angle. Since 315^∘ is in Quadrant IV, cos(315^∘)=cos(45^∘)

cos(315^∘)=√(,2)/2

5. Substitute the value back into the formula.

cos(157.5^∘)=−√(,(1+√(,2)/2)/2)

6. Simplify the expression inside the radical by finding a common denominator in the numerator.

cos(157.5^∘)=−√(,(2+√(,2))/2/2)

7. Divide by 2 to reach the final simplified form.

cos(157.5^∘)=−√(,(2+√(,2))/4)

8. Simplify the square root of the denominator.

cos(157.5^∘)=−√(,2+√(,2))/2

Final Answer

cos(157.5^∘)=−√(,2+√(,2))/2

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