Find the Value Using the Unit Circle cos((3pi)/8)

Problem

cos((3*π)/8)

Solution

1. Identify the angle (3*π)/8 as half of a known angle from the unit circle, specifically (3*π)/4

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

3. Substitute θ=(3*π)/4 into the formula.

4. Determine the sign based on the quadrant. Since (3*π)/8 is in the first quadrant, the cosine value must be positive.

5. Evaluate cos((3*π)/4) using the unit circle, which is −√(,2)/2

6. Substitute the value into the radical expression:

cos((3*π)/8)=√(,(1+(−√(,2)/2))/2)

7. Simplify the numerator by finding a common denominator:

cos((3*π)/8)=√(,(2−√(,2))/2/2)

8. Divide by 2 to simplify the fraction inside the square root:

cos((3*π)/8)=√(,(2−√(,2))/4)

9. Simplify the radical by taking the square root of the denominator:

cos((3*π)/8)=√(,2−√(,2))/2

Final Answer

cos((3*π)/8)=√(,2−√(,2))/2

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