Simplify Using Half-Angle Formula cos(15 degrees )

Problem

cos(15)

Solution

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

2. Determine the value of θ by setting θ/2=15 which gives θ=30

3. Choose the positive sign for the square root because 15 is in the first quadrant, where cosine is positive.

4. Substitute θ=30 into the formula:

cos(15)=√(,(1+cos(30))/2)

5. Evaluate cos(30) using the unit circle value √(,3)/2

cos(15)=√(,(1+√(,3)/2)/2)

6. Simplify the numerator by finding a common denominator:

cos(15)=√(,(2+√(,3))/2/2)

7. Divide the fraction by 2:

cos(15)=√(,(2+√(,3))/4)

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

cos(15)=√(,2+√(,3))/2

Final Answer

cos(15)=√(,2+√(,3))/2

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