Find the Exact Value cos(15)

Problem

cos(15^∘)

Solution

1. Identify the angle as a difference of two special angles from the unit circle, such as 45^∘ and 30^∘

2. Apply the formula for the cosine of a difference, which is cos(A−B)=cos(A)*cos(B)+sin(A)*sin(B)

3. Substitute A=45^∘ and B=30^∘ into the identity.

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

4. Evaluate the trigonometric functions using exact values: cos(45^∘)=√(,2)/2 cos(30^∘)=√(,3)/2 sin(45^∘)=√(,2)/2 and sin(30^∘)=1/2

cos(15^∘)=√(,2)/2⋅√(,3)/2+√(,2)/2⋅1/2

5. Simplify the expression by multiplying the fractions and combining the terms over a common denominator.

cos(15^∘)=√(,6)/4+√(,2)/4

cos(15^∘)=(√(,6)+√(,2))/4

Final Answer

cos(15^∘)=(√(,6)+√(,2))/4

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