Simplify cos(15)

Problem

cos(15)

Solution

1. Rewrite the angle 15 as a difference of two special angles whose trigonometric values are known, such as 45 and 30

cos(15)=cos(45−30)

2. Apply the cosine difference formula, which states cos(α−β)=cos(α)*cos(β)+sin(α)*sin(β)

cos(45−30)=cos(45)*cos(30)+sin(45)*sin(30)

3. Substitute the exact values for the sine and cosine of 45 and 30

cos(45)=√(,2)/2

cos(30)=√(,3)/2

sin(45)=√(,2)/2

sin(30)=1/2

4. Multiply the terms together.

cos(15)=√(,2)/2⋅√(,3)/2+√(,2)/2⋅1/2

cos(15)=√(,6)/4+√(,2)/4

5. Combine the fractions over a common denominator.

cos(15)=(√(,6)+√(,2))/4

Final Answer

cos(15)=(√(,6)+√(,2))/4

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