Find the Exact Value sin((2pi)/3+(3pi)/4)

Problem

sin((2*π)/3+(3*π)/4)

Solution

1. Identify the sum formula for sine, which is sin(A+B)=sin(A)*cos(B)+cos(A)*sin(B)

2. Assign the values A=(2*π)/3 and B=(3*π)/4 to the formula.

3. Substitute the values into the identity:

sin((2*π)/3+(3*π)/4)=sin((2*π)/3)*cos((3*π)/4)+cos((2*π)/3)*sin((3*π)/4)

4. Evaluate the trigonometric functions for the given angles:

sin((2*π)/3)=√(,3)/2

cos((3*π)/4)=−√(,2)/2

cos((2*π)/3)=−1/2

sin((3*π)/4)=√(,2)/2

5. Multiply the evaluated terms:

(√(,3)/2)*(−√(,2)/2)+(−1/2)*(√(,2)/2)

−√(,6)/4−√(,2)/4

6. Combine the fractions over a common denominator.

Final Answer

sin((2*π)/3+(3*π)/4)=(−√(,6)−√(,2))/4

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