Let's analyze the question step by step. The sine function is generally written as:
$y = A \sin(B(x - C))$
where:

• $A$ is the amplitude.
• $B$ determines the period, with $\text{Period} = \frac{2\pi}{B}$.
• $C$ is the phase shift.

Given information:

• Amplitude ($A$) = 3.
• Period = $7\pi$.
• Phase shift ($C$) = $\frac{1}{6}$.

Step 1: Find $B$

The period formula is: $\text{Period} = \frac{2\pi}{B}.$ Substitute the period $7\pi$: $7\pi = \frac{2\pi}{B}.$ Solve for $B$: $B = \frac{2\pi}{7\pi} = \frac{2}{7}.$

Step 2: Use phase shift ($C$)

The phase shift $C = \frac{1}{6}$. This represents a horizontal shift of the graph.

Final Equation:

Now substitute into the general form: $y = 3 \sin\left(\frac{2}{7}\left(x - \frac{1}{6}\right)\right).$

Simplify to match the provided blanks: $y = 3 \sin\left(\frac{2}{7}x - \frac{2}{42}\right) = 3 \sin\left(\frac{2}{7}x - \frac{1}{21}\right).$

Thus, the values to fill in are:

• First blank: $\frac{2}{7}$.
• Second blank: $\frac{1}{21}$.

Let me know if you'd like a deeper explanation!

Related Questions:

1. How does changing $A$ affect the sine graph?
2. What is the physical interpretation of phase shift in sine functions?
3. How do you calculate the period for cosine functions?
4. How does changing $B$ influence the graph's frequency?
5. Can you derive the equation for a cosine function with the same characteristics?

Tip:

When simplifying fractions in trigonometric functions, always check for the lowest terms to match standard forms!