Find the Exact Value cos(15 degrees )

Problem

cos(15^∘)

Solution

1. Rewrite the angle 15^∘ as a difference of two common angles from the unit circle, such as 45^∘−30^∘

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

3. Substitute the values α=45^∘ and β=30^∘ into the formula.

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. Multiply the fractions in each term.

cos(15^∘)=√(,6)/4+√(,2)/4

6. Combine the terms over a common denominator.

cos(15^∘)=(√(,6)+√(,2))/4

Final Answer

cos(15^∘)=(√(,6)+√(,2))/4

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