Find the Exact Value cos(pi/16)

Problem

cos(π/16)

Solution

1. Identify the half-angle identity for cosine, which is cos(θ/2)=√(,(1+cos(θ))/2) for angles in the first quadrant.

2. Apply the formula to θ=π/4 to find cos(π/8)

cos(π/8)=√(,(1+cos(π/4))/2)

3. Substitute the known value cos(π/4)=√(,2)/2 into the expression.

cos(π/8)=√(,(1+√(,2)/2)/2)

4. Simplify the fraction inside the square root.

cos(π/8)=√(,(2+√(,2))/4)

cos(π/8)=√(,2+√(,2))/2

5. Apply the formula again using θ=π/8 to find cos(π/16)

cos(π/16)=√(,(1+cos(π/8))/2)

6. Substitute the value of cos(π/8) found in step 4.

cos(π/16)=√(,(1+√(,2+√(,2))/2)/2)

7. Simplify the final expression by combining the terms in the numerator and dividing by 2.

cos(π/16)=√(,(2+√(,2+√(,2)))/4)

cos(π/16)=√(,2+√(,2+√(,2)))/2

Final Answer

cos(π/16)=√(,2+√(,2+√(,2)))/2

Want more problems? Check here!