We are tasked with finding the equations of the tangent line and the normal line to the function $f(x) = 3 \sin(x)$ at $x = \frac{\pi}{2}$.

Step 1: Find the value of $f(x)$ at $x = \frac{\pi}{2}$

We first evaluate the function $f(x)$ at $x = \frac{\pi}{2}$: $f\left(\frac{\pi}{2}\right) = 3 \sin\left(\frac{\pi}{2}\right) = 3 \times 1 = 3$ So, the point on the graph is $\left( \frac{\pi}{2}, 3 \right)$.

Step 2: Find the derivative of $f(x)$ (the slope of the tangent line)

To get the slope of the tangent line, we differentiate $f(x) = 3 \sin(x)$: $f'(x) = 3 \cos(x)$ Now, substitute $x = \frac{\pi}{2}$ into the derivative: $f'\left(\frac{\pi}{2}\right) = 3 \cos\left(\frac{\pi}{2}\right) = 3 \times 0 = 0$ The slope of the tangent line at $x = \frac{\pi}{2}$ is 0.

Step 3: Equation of the tangent line

The general form of the equation of a line is: $y - y_1 = m(x - x_1)$ Here, $m = 0$, $x_1 = \frac{\pi}{2}$, and $y_1 = 3$. Substituting these values: $y - 3 = 0(x - \frac{\pi}{2})$ $y = 3$ Thus, the equation of the tangent line is: $\boxed{y = 3}$

Step 4: Slope of the normal line

The slope of the normal line is the negative reciprocal of the slope of the tangent line. Since the slope of the tangent line is 0, the slope of the normal line will be undefined, meaning the normal line is a vertical line.

Step 5: Equation of the normal line

Since the normal line is vertical and passes through $x = \frac{\pi}{2}$, the equation of the normal line is: $\boxed{x = \frac{\pi}{2}}$

Final Answer:

• The equation of the tangent line is $y = 3$.
• The equation of the normal line is $x = \frac{\pi}{2}$.

Would you like further details or have any questions?

Here are 5 related questions for practice:

1. Find the tangent and normal lines to $f(x) = \sin(x)$ at $x = \frac{\pi}{3}$.
2. Find the equation of the tangent line to $f(x) = \cos(x)$ at $x = \frac{\pi}{4}$.
3. Determine the slope of the tangent line to $f(x) = 2\sin(x) + x^2$ at $x = 0$.
4. What is the equation of the tangent line to $f(x) = e^x \sin(x)$ at $x = 0$?
5. Find the point where the tangent line to $f(x) = 3x - 2 \sin(x)$ is horizontal.

Tip: The slope of a normal line is always the negative reciprocal of the slope of the tangent line at the same point.