Simplify

Problem

sin(π/12)*cos((7*π)/12)+cos(π/12)*sin((7*π)/12)

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 sine sum formula, which states:

sin(A+B)=sin(A)*cos(B)+cos(A)*sin(B)

3. Substitute the values A=π/12 and B=(7*π)/12 into the formula:

sin(π/12+(7*π)/12)

4. Add the fractions inside the sine function:

π/12+(7*π)/12=(8*π)/12

5. Simplify the fraction by dividing the numerator and denominator by their greatest common divisor, 4:

(8*π)/12=(2*π)/3

6. Evaluate the sine of the resulting angle using the unit circle:

sin((2*π)/3)=√(,3)/2

Final Answer

sin(π/12)*cos((7*π)/12)+cos(π/12)*sin((7*π)/12)=√(,3)/2

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