Simplify

Problem

sin((5*π)/12)*cos(π/3)+cos((5*π)/12)*sin(π/3)

Solution

1. Identify the trigonometric identity that matches the structure of the expression. The expression follows the form sin(A)*cos(B)+cos(A)*sin(B)

2. Apply the formula for the sine of a sum, which states sin(A+B)=sin(A)*cos(B)+cos(A)*sin(B)

3. Substitute the values A=(5*π)/12 and B=π/3 into the sum formula.

sin((5*π)/12+π/3)

4. Find a common denominator to add the fractions inside the sine function. The common denominator for 12 and 3 is 12

π/3=(4*π)/12

5. Add the angles together.

(5*π)/12+(4*π)/12=(9*π)/12

6. Simplify the resulting fraction by dividing the numerator and denominator by 3

(9*π)/12=(3*π)/4

7. Evaluate the sine of the simplified angle using the unit circle.

sin((3*π)/4)=√(,2)/2

Final Answer

sin((5*π)/12)*cos(π/3)+cos((5*π)/12)*sin(π/3)=√(,2)/2

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