Questions: Make a sine wave with the given amplitude, period or frequency, and phase shift. Use the form y=A sin (B(t-C)). Assume A>0. Amplitude 2, period π/3, and phase shift π/3 (Express numbers in exact form. Use symbolic notation and fractions where needed. Give the equations in terms of t and y, where t is the independent variable and y is the dependent variable.) equation: 2 sin (6(t-π/3)) Sketch the graph over two periods.

Make a sine wave with the given amplitude, period or frequency, and phase shift. Use the form y=A sin (B(t-C)). Assume A>0. Amplitude 2, period π/3, and phase shift π/3 (Express numbers in exact form. Use symbolic notation and fractions where needed. Give the equations in terms of t and y, where t is the independent variable and y is the dependent variable.) equation: 2 sin (6(t-π/3)) Sketch the graph over two periods.
Transcript text: Make a sine wave with the given amplitude, period or frequency, and phase shift. Use the form $y=A \sin (B(t-C))$. Assume $A>0$. Amplitude 2 , period $\frac{\pi}{3}$, and phase shift $\frac{\pi}{3}$ (Express numbers in exact form. Use symbolic notation and fractions where needed. Give the equations in terms of $t$ and $y$, where $t$ is the independent variable and $y$ is the dependent variable.) equation: $2 \sin \left(6\left(t-\frac{\pi}{3}\right)\right)$ Sketch the graph over two periods.
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Solution

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Solution Steps

Step 1: Find B

The period is given as π/3. The relationship between the period and B is given by Period = 2π/B. So, π/3 = 2π/B. Solving for B gives B = 6.

Step 2: Write the equation

We are given that the amplitude A = 2, the phase shift C = π/3, and we found B = 6. Plugging these values into the general form y = A sin(B(t - C)) yields:

y = 2sin(6(t - π/3)).

Step 3: Sketch the graph

Since the period is π/3, two periods will be 2π/3. The sine wave starts at t=π/3 and the first period ends at t= π/3 + π/3 = 2π/3. The second period ends at t= 2π/3 + π/3 = π.

The amplitude is 2, so the graph will oscillate between y = -2 and y = 2. The graph crosses the t-axis at t=π/3, t=2π/3, and t = π. It reaches its maximum at t = 7π/12 (between π/3 and 2π/3, with a value of 2) It reaches its minimum at t = π/12 (between 2π/3 and π, with a value of -2) (Note: the grid lines are spaced π/12)

Final Answer:

The equation is y = 2sin(6(t - π/3)). The graph is a sine wave with amplitude 2, period π/3, and phase shift π/3. It starts at (π/3, 0), reaches a maximum of 2 at t = π/3 + π/12, crosses the t-axis again at t= 2π/3, reaches a minimum of -2 at t = 2π/3 + π/12, and crosses the t-axis again at t = π.

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