Determine the exact value of the expression without the use of a calculator.

Question

sin(π/12) - sin(17π/12)

 

(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Answer 1

Rewrite the angles so that they are a sum of difference of angles on the unit circle.

 

π/12 = (4π/12) - (2π/12) - (π/12)

        = (3π/12) - (2π/12)

        = π/4 - π/6

 

17π/12 = (8π/12) + (9π/12)

           = (2π/3) + (3π/4)

 

 

Use the subtraction angle identity.

 

sin(π/4 - π/6) = sin(π/4)cos(π/6) - cos(π/4)sin(π/6)

                     = (√(2)/2)(√(3)/2 - 1/2)

                     = √(6)/4 - √(3)/4

 

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

                            = (√(3)/2)(-√(2)/2) + (1/2)(√(2)/2)

                            = -√(6)/4 + √(2)/ 4

 

 

Subtract the bolded terms together to get your difference.

Answer 2

Here's a formula that might be useful:  sin w - sin z = 2 cos((w+z)/2) sin((w-z)/2)