Problem 30 - Precalculus Essentials

Question

Write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. $$\sin \frac{7 \pi}{12} \cos \frac{\pi}{12}-\cos \frac{7 \pi}{12} \sin \frac{\pi}{12}$$

Step-by-Step Solution

Answer

The exact value of the expression is 1.

Step 1: Identify the Difference of Sine Property

Analyze the given expression, here \(\sin \frac{7 \pi}{12} \cos \frac{\pi}{12}-\cos \frac{7 \pi}{12} \sin \frac{\pi}{12}\), it can be noticed that it's in the form of the difference of sine formula. This can be written as \(\sin (A-B)\), where \(A = \frac{7 \pi}{12}\) and \(B = \frac{\pi}{12}\).

Step 2: Substitution of the Values

Substitute those values into the formula to get the expression \(\sin \left(\frac{7 \pi}{12} - \frac{\pi}{12}\right)\). Simplify it further, we get the expression \(\sin \left(\frac{6 \pi}{12}\right)\).

Step 3: Solve the Trigonometric Expression

The simplified expression equals to \(\sin(\frac{\pi}{2})\), as \(\frac{6 \pi}{12}\) simplifies to \(\frac{\pi}{2}\). The sine of \(\frac{\pi}{2}\) radians or 90 degrees is 1.