Simplify cos(-15)

Problem

cos(−15)

Solution

1. Apply the even-odd identity for the cosine function, which states cos(−θ)=cos(θ)

cos(−15)=cos(15)

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

cos(15)=cos(45−30)

3. Apply the cosine difference formula, cos(A−B)=cos(A)*cos(B)+sin(A)*sin(B) where A=45 and B=30

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

4. Substitute the exact values for the trigonometric functions of 45 and 30

cos(45−30)=(√(,2)/2)*(√(,3)/2)+(√(,2)/2)*(1/2)

5. Multiply the fractions in each term.

cos(45−30)=√(,6)/4+√(,2)/4

6. Combine the terms over a common denominator.

cos(45−30)=(√(,6)+√(,2))/4

Final Answer

cos(−15)=(√(,6)+√(,2))/4

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