Question

write the expression as the sine, cosine, or tangent of an angle. then find the exact va

sin \frac { 5 pi } { 12 } cos \frac { pi } { 4 } - cos \frac { 5 pi } { 12 } sin \frac { pi } { 4 }

write the expression as the sine, cosine, or tangent of a single angle.

sin \frac { 5 pi } { 12 } cos \frac { pi } { 4 } - cos \frac { 5 pi } { 12 } sin \frac { pi } { 4 } =
(use integers or fractions for any numbers in the expression. simplify your answer. do

Explanation:

Step1: Use the sine subtraction formula

The formula for \(\sin(A - B)=\sin A\cos B-\cos A\sin B\). Here \(A = \frac{5\pi}{12}\) and \(B=\frac{\pi}{4}\). So, \(\sin\frac{5\pi}{12}\cos\frac{\pi}{4}-\cos\frac{5\pi}{12}\sin\frac{\pi}{4}=\sin(\frac{5\pi}{12}-\frac{\pi}{4})\).

Step2: Simplify the angle

Calculate \(\frac{5\pi}{12}-\frac{\pi}{4}=\frac{5\pi}{12}-\frac{3\pi}{12}=\frac{5\pi - 3\pi}{12}=\frac{\pi}{6}\).

Answer:

\(\sin\frac{\pi}{6}\)