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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