Problem 34 - Precalculus Essentials

Question

Verify each identity. $$\sin \left(x+\frac{3 \pi}{2}\right)=-\cos x$$

Step-by-Step Solution

Answer

The identity is correct.

Step 1: Recognize the Sum Identity for Sine

The Sum Identity for sine states that for any two angles \(A\) and \(B\), \(\sin(A + B) = \sin A \cos B + \cos A \sin B\). We can apply this to our equation by letting \(A = x\) and \(B = \frac{3\pi}{2}\) . Thus, \(\sin \left(x + \frac{3 \pi}{2}\right) = \sin x \cos \frac{3 \pi}{2} + \cos x \sin \frac{3 \pi}{2}\).

Step 2: Evaluate the Trigonometric Functions

We know that \(\cos \frac{3\pi}{2} = 0\) and \(\sin \frac{3\pi}{2} = -1\). Substituting these into the equation gives us the result \(= \sin x * 0 + \cos x * (-1)\). This simplifies to \(-\cos x\).

Step 3: Compare both Sides

We can now clearly see that both sides of our original equation are equal: \(\sin \left(x + \frac{3 \pi}{2}\right) = -\cos x\). Therefore, we have verified the given identity.