Find all points on the graph of the function 
f(x) = 2 sin(x) + sin2(x)
 at which the tangent line is horizontal. (Use n as your arbitrary integer.)

To find all points where the tangent line to the graph of \( f(x) = 2\sin(x) + \sin^2(x) \) is horizontal, we need to find the values of \( x \) where the derivative of the function, \( f'(x) \), equals zero.

### Step 1: Differentiate the function

The given function is:
\[
f(x) = 2\sin(x) + \sin^2(x)
\]

Now, differentiate \( f(x) \) with respect to \( x \):

1. The derivative of \( 2\sin(x) \) is:
   \[
   \frac{d}{dx}[2\sin(x)] = 2\cos(x)
   \]

2. The derivative of \( \sin^2(x) \) (using the chain rule) is:
   \[
   \frac{d}{dx}[\sin^2(x)] = 2\sin(x)\cos(x)
   \]

Thus, the derivative of \( f(x) \) is:
\[
f'(x) = 2\cos(x) + 2\sin(x)\cos(x)
\]

### Step 2: Set the derivative equal to zero

To find where the tangent line is horizontal, set \( f'(x) = 0 \):

\[
2\cos(x) + 2\sin(x)\cos(x) = 0
\]

Factor out \( 2\cos(x) \):

\[
2\cos(x)[1 + \sin(x)] = 0
\]

### Step 3: Solve for \( x \)

For this product to equal zero, either \( \cos(x) = 0 \) or \( 1 + \sin(x) = 0 \).

#### Case 1: \( \cos(x) = 0 \)

The solutions to \( \cos(x) = 0 \) are:
\[
x = \frac{\pi}{2} + n\pi, \quad n \in \mathbb{Z}
\]
where \( n \) is any integer.

#### Case 2: \( 1 + \sin(x) = 0 \)

Solve for \( \sin(x) \):
\[
\sin(x) = -1
\]
The solutions to \( \sin(x) = -1 \) are:
\[
x = \frac{3\pi}{2} + 2n\pi, \quad n \in \mathbb{Z}
\]

### Step 4: Combine the solutions

Thus, the points at which the tangent line is horizontal are given by:
\[
x = \frac{\pi}{2} + n\pi \quad \text{or} \quad x = \frac{3\pi}{2} + 2n\pi
\]
where \( n \in \mathbb{Z} \).

These are all the \( x \)-coordinates where the tangent line is horizontal. To find the corresponding points on the graph, substitute these values of \( x \) into the original function \( f(x) \).

---

Would you like details on finding the actual points on the graph or explanations of any steps?

Here are 5 related questions:
1. How do you find the derivative of a trigonometric function?
2. What is the significance of the horizontal tangent line on a graph?
3. How would you solve other equations involving trigonometric identities?
4. Can you find the horizontal tangent points for the function \( g(x) = 3\sin(x) + \cos^2(x) \)?
5. What is the graphical interpretation of solving \( \cos(x) = 0 \)?

**Tip:** Always check for factoring opportunities when solving equations involving products, as this can simplify solving significantly.

Find all points on the graph of the function f(x) = 2 sin(x) + sin^2(x) at which the tangent line is horizontal. (Use n as your arbitrary integer.)

Discover how to find all points on the graph of the function f(x) = 2sin(x) + sin^2(x) where the tangent line is horizontal. This problem involves differentiating trigonometric functions, using the chain rule, and solving equations like cos(x) = 0. Suitable for Grades 11-12 calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation