Find the Exact Value cos(pi/4+pi/3)

Problem

cos(π/4+π/3)

Solution

1. Identify the appropriate trigonometric identity for the cosine of a sum, which is cos(A+B)=cos(A)*cos(B)−sin(A)*sin(B)

2. Substitute the values A=π/4 and B=π/3 into the identity.

cos(π/4+π/3)=cos(π/4)*cos(π/3)−sin(π/4)*sin(π/3)

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

cos(π/4)=√(,2)/2

cos(π/3)=1/2

sin(π/4)=√(,2)/2

sin(π/3)=√(,3)/2

4. Multiply the fractions together.

√(,2)/2⋅1/2=√(,2)/4

√(,2)/2⋅√(,3)/2=√(,6)/4

5. Subtract the resulting terms to find the final exact value.

√(,2)/4−√(,6)/4=(√(,2)−√(,6))/4

Final Answer

cos(π/4+π/3)=(√(,2)−√(,6))/4

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