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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