Find Amplitude, Period, and Phase Shift y=10sin(theta/6+30)-5

Solution:

Step 1:

Identify the parameters $a$, $b$, $c$, and $d$ from the standard trigonometric form $a\sin (b\theta +c)+d$ to determine amplitude, period, phase shift, and vertical shift.

• Amplitude coefficient $a=10$
• Frequency coefficient $b=\frac{1}{6}$
• Phase shift constant $c=30$ (note the sign change due to the form used)
• Vertical shift $d=-5$

Step 2:

Calculate the amplitude as the absolute value of $a$.

• Amplitude: $|a|=|10|=10$

Step 3:

Compute the period using the formula $T=\frac{2\pi }{|b|}$.

Step 3.1:

Determine the period for the sine function $10\sin (\frac{\theta }{6}+30)$.

Step 3.1.1:

Use the period formula $T=\frac{2\pi }{|b|}$.

Step 3.1.2:

Substitute $b$ with $\frac{1}{6}$ to find the period.

• $T=\frac{2\pi }{|\frac{1}{6}|}$

Step 3.1.3:

Since $\frac{1}{6}$ is positive, we can disregard the absolute value.

• $T=\frac{2\pi }{\frac{1}{6}}$

Step 3.1.4:

Calculate the period by multiplying $2\pi$ by the reciprocal of $\frac{1}{6}$.

• $T=2\pi \cdot 6$

Step 3.1.5:

Simplify the multiplication to get the period.

• Period: $T=12\pi$

Step 3.2:

The constant term $-5$ does not affect the period of the sine function.

Step 4:

Determine the phase shift using the formula $\text{Phase Shift}=\frac{c}{b}$.

Step 4.1:

Calculate the phase shift with $\text{Phase Shift}=\frac{c}{b}$.

Step 4.2:

Insert the values for $c$ and $b$ into the phase shift equation.

• Phase Shift: $\frac{30}{\frac{1}{6}}$

Step 4.3:

Multiply the numerator by the reciprocal of the denominator to find the phase shift.

• Phase Shift: $30\cdot 6$

Step 4.4:

Complete the multiplication to obtain the phase shift.

• Phase Shift: $180$ (This is a shift to the right, opposite to the original solution's direction)

Step 5:

Summarize the properties of the trigonometric function.

• Amplitude: $10$
• Period: $12\pi$
• Phase Shift: $180$ (to the right)
• Vertical Shift: $-5$

Knowledge Notes:

The amplitude of a trigonometric function is the coefficient in front of the sine or cosine function, which determines the height of the wave's peaks and troughs.

The period of a sine or cosine function is the length of one complete cycle of the wave, which can be calculated using the formula $T=\frac{2\pi }{|b|}$, where $b$ is the frequency coefficient.

The phase shift of a trigonometric function is the horizontal shift along the x-axis and is calculated by the formula $\text{Phase Shift}=\frac{c}{b}$, where $c$ is the phase shift constant in the function's equation.

The vertical shift is the constant $d$ added or subtracted from the trigonometric function, which moves the graph up or down along the y-axis.

In the context of the given problem, the trigonometric function is a sine function with modifications to its amplitude, period, phase shift, and vertical shift due to the coefficients and constants present in the equation $y=10\sin (\frac{\theta }{6}+30)-5$.