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

Problem

cos(π/3+π/4)

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/3 and B=π/4 into the identity.

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

3. Determine the exact values of the trigonometric functions for the given angles from the unit circle.

cos(π/3)=1/2

cos(π/4)=√(,2)/2

sin(π/3)=√(,3)/2

sin(π/4)=√(,2)/2

4. Multiply the substituted values together.

cos(π/3+π/4)=(1/2)*(√(,2)/2)−(√(,3)/2)*(√(,2)/2)

cos(π/3+π/4)=√(,2)/4−√(,6)/4

5. Combine the fractions over a common denominator.

cos(π/3+π/4)=(√(,2)−√(,6))/4

Final Answer

cos(π/3+π/4)=(√(,2)−√(,6))/4

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