Simplify cos(45-30)

Problem

cos(45−30)

Solution

1. Identify the appropriate trigonometric identity for the cosine of a difference, which is cos(α−β)=cos(α)*cos(β)+sin(α)*sin(β)

2. Substitute the values α=45 and β=30 into the identity.

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

3. Evaluate the trigonometric functions for the known special angles.

cos(45)=√(,2)/2

cos(30)=√(,3)/2

sin(45)=√(,2)/2

sin(30)=1/2

4. Multiply the terms together.

√(,2)/2⋅√(,3)/2=√(,6)/4

√(,2)/2⋅1/2=√(,2)/4

5. Combine the fractions over a common denominator.

√(,6)/4+√(,2)/4=(√(,6)+√(,2))/4

Final Answer

cos(45−30)=(√(,6)+√(,2))/4

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