Math / Trigonometry

QuestionQuestion 6 (1 point) Saved
Determine the phase shift of the sinusoidal function $y=-5 \sin \left(3 x-\frac{3 \pi}{2}\right)-5$ $\frac{3 \pi}{2} \mathrm{rad}$ $\frac{\pi}{2} \mathrm{rad}$ 0.5 rad to the right 0.5 rad to the left
Question 7 (1 point) $\checkmark$ Saved
An equivalent trigonometric expression for $\sin \left(\frac{\pi}{2}-x\right)$ is $\sin x$ $\tan x$ $\cos x$ none of the above
Question 8 (1 point) $\checkmark$ Saved
An equivalent trigonometric expression for $\cos (-x)$ is $\sin x$ $-\sin x$ $\cos x$

Studdy Solution

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:
$\sin \left(\frac{\pi}{2} - x\right) = \cos x$

STEP 7

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 $\cos x$ .

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Studdy Solution

STEP 1

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:
$\sin \left(\frac{\pi}{2} - x\right) = \cos x$

STEP 7

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Studdy Solution

STEP 1

STEP 2

1. Analyze the sinusoidal function to find the phase shift.2. Use trigonometric identities to find an equivalent expression for sin⁡(π2−x) \sin \left(\frac{\pi}{2} - x\right) sin(2π​−x).3. Use trigonometric identities to find an equivalent expression for cos⁡(−x) \cos(-x) cos(−x).

STEP 3

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

STEP 4

Identify b b b and c c c from the function y=−5sin⁡(3x−3π2)−5 y = -5 \sin \left(3x - \frac{3\pi}{2}\right) - 5 y=−5sin(3x−23π​)−5:Here, b=3 b = 3 b=3 and c=3π2 c = \frac{3\pi}{2} c=23π​.

STEP 5

Calculate the phase shift:Phase shift=cb=3π23=π2 \text{Phase shift} = \frac{c}{b} = \frac{\frac{3\pi}{2}}{3} = \frac{\pi}{2} Phase shift=bc​=323π​​=2π​Since the expression is bx−c bx - c bx−c, the phase shift is to the right.

STEP 6

Use the co-function identity for sine:sin⁡(π2−x)=cos⁡x \sin \left(\frac{\pi}{2} - x\right) = \cos x sin(2π​−x)=cosx

STEP 7

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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 $\cos(-x)$ .

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

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

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.

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