Math Snap

STEP 1

1. We are given a sinusoidal function $y = -5 \sin \left(3x - \frac{3\pi}{2}\right) - 5$.
2. We need to determine the phase shift of this function.
3. We are given a trigonometric expression $\sin \left(\frac{\pi}{2} - x\right)$ and need to find its equivalent.
4. We are given a trigonometric expression $\cos(-x)$ and need to find its equivalent.

STEP 2

1. Analyze the sinusoidal function to find the phase shift.
2. Use trigonometric identities to find an equivalent expression for $\sin \left(\frac{\pi}{2} - x\right)$.
3. Use trigonometric identities to find an equivalent expression for $\cos(-x)$.

STEP 3

The general form of a sinusoidal function is $y = a \sin(bx - c) + d$.
The phase shift is given by $\frac{c}{b}$.

STEP 4

Identify $b$ and $c$ from the function $y = -5 \sin \left(3x - \frac{3\pi}{2}\right) - 5$:
Here, $b = 3$ and $c = \frac{3\pi}{2}$.

STEP 5

Calculate the phase shift:
$$\text{Phase shift} = \frac{c}{b} = \frac{\frac{3\pi}{2}}{3} = \frac{\pi}{2}$$ Since the expression is $bx - c$, the phase shift is to the right.

STEP 6

Use the co-function identity for sine:
$$\sin \left(\frac{\pi}{2} - x\right) = \cos x$$

Use the even-odd identity for cosine:
$$\cos(-x) = \cos x$$ The answers are:
1. The phase shift of the sinusoidal function is $\frac{\pi}{2}$ radians to the right.
2. An equivalent trigonometric expression for $\sin \left(\frac{\pi}{2} - x\right)$ is $\cos x$.
3. An equivalent trigonometric expression for $\cos(-x)$ is $\cos x$.