Precalc Trigonometric Function

Question

Use an Addition or Subtraction Formula to write the expression as a trigonometric function of one number.

Sin (4𝜋/7) Cos(5𝜋/21) -Cos(4𝜋/7) Sin(5𝜋/21)

A.  Write Expression as a trigonometric function of one number.

B. Find it's exact value

Answer 1

The angle subtraction identity for sine looks like this:

which means we can apply it to your case as follows:

sin(4π/7 - 5π/21) = sin(4π/7)cos(5π/21) - cos(4π/7)sin(5π/21)

Getting a common denomination we can re-write sin(4π/7 - 5π/21) as sin(12π/21 - 5π/21) and simplify to sin(7π/21) = sin(π/3)

Using the unit circle, we know that sin(π/3) = √3/2