Find the Exact Value 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 exact trigonometric values for the given angles.

cos(45)=√(,2)/2

cos(30)=√(,3)/2

sin(45)=√(,2)/2

sin(30)=1/2

4. Multiply the terms together.

cos(45−30)=√(,2)/2⋅√(,3)/2+√(,2)/2⋅1/2

cos(45−30)=√(,6)/4+√(,2)/4

5. Combine the fractions over a common denominator.

cos(45−30)=(√(,6)+√(,2))/4

Final Answer

cos(45−30)=(√(,6)+√(,2))/4

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