Expand Using Sum/Difference Formulas cos(15)

Problem

cos(15)

Solution

1. Identify two angles with known trigonometric values whose difference is 15 We can use 45 and 30

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

3. Substitute 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 the unit circle values.

cos(45)=√(,2)/2

cos(30)=√(,3)/2

sin(45)=√(,2)/2

sin(30)=1/2

5. Multiply the terms together.

√(,2)/2⋅√(,3)/2=√(,6)/4

√(,2)/2⋅1/2=√(,2)/4

6. Combine the fractions over a common denominator.

(√(,6)+√(,2))/4

Final Answer

cos(15)=(√(,6)+√(,2))/4

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