Problem 29 - Precalculus Essentials

Question

Write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. $$\sin \frac{5 \pi}{12} \cos \frac{\pi}{4}-\cos \frac{5 \pi}{12} \sin \frac{\pi}{4}$$

Step-by-Step Solution

Answer

The exact value of the given expression is \(\sqrt{3}/2\)

Step 1: Identify the expression type

In this expression, we see that its format is similar to the sine difference identity: \(\sin(A - B) = \sin A \cos B - \cos A \sin B\). So we can recognize this as the sine of the difference between two angles.

Step 2: Apply the identity

Using the sine difference identity, our given expression \(\sin \frac{5 \pi}{12} \cos \frac{\pi}{4}-\cos \frac{5 \pi}{12} \sin \frac{\pi}{4}\) can be collapsed into \(\sin(\frac{5 \pi}{12}-\frac{\pi}{4})\)

Step 3: Simplify the difference in the sine function

Let's simplify the difference inside the sine function, \(\frac{5 \pi}{12}-\frac{\pi}{4}\) = \(\frac{\pi}{3}\)

Step 4: Calculate the value

We know that \(\sin \frac{\pi}{3} = \sqrt{3}/2\), as it's one of the special angles in trigonometric functions, we use the simplified angle to find the value of the expression