Find the equation of the line tangent to the graph of f(θ)= tanθ sinθ when θ= π/4. Use the quotient rule

To find the equation of the tangent line to the graph of \( f(\theta) = \tan \theta \sin \theta \) at \( \theta = \frac{\pi}{4} \), we need to follow these steps:

1. **Differentiate** \( f(\theta) = \tan \theta \sin \theta \) using the product rule (or quotient rule, if rewritten as a quotient).
2. **Evaluate** \( f'(\theta) \) at \( \theta = \frac{\pi}{4} \).
3. **Find the point on the curve** at \( \theta = \frac{\pi}{4} \).
4. **Construct the tangent line** using the point-slope form of a line.

### Step 1: Differentiate \( f(\theta) = \tan \theta \sin \theta \)

We can use the product rule for differentiation:
\[
f'(\theta) = \frac{d}{d\theta} (\tan \theta) \cdot \sin \theta + \tan \theta \cdot \frac{d}{d\theta} (\sin \theta)
\]

The derivatives are:
\[
\frac{d}{d\theta} (\tan \theta) = \sec^2 \theta \quad \text{and} \quad \frac{d}{d\theta} (\sin \theta) = \cos \theta
\]

Thus,
\[
f'(\theta) = (\sec^2 \theta) \sin \theta + (\tan \theta) \cos \theta
\]

### Step 2: Evaluate \( f'(\theta) \) at \( \theta = \frac{\pi}{4} \)

At \( \theta = \frac{\pi}{4} \):
- \( \tan \frac{\pi}{4} = 1 \)
- \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \)
- \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \)
- \( \sec^2 \frac{\pi}{4} = 2 \) (since \( \sec \frac{\pi}{4} = \sqrt{2} \))

Plugging these values into \( f'(\theta) \):
\[
f'\left( \frac{\pi}{4} \right) = (2) \cdot \frac{\sqrt{2}}{2} + (1) \cdot \frac{\sqrt{2}}{2} = \sqrt{2} + \frac{\sqrt{2}}{2} = \frac{3\sqrt{2}}{2}
\]

### Step 3: Find the point on the curve at \( \theta = \frac{\pi}{4} \)

To find \( f\left( \frac{\pi}{4} \right) \):
\[
f\left( \frac{\pi}{4} \right) = \tan \frac{\pi}{4} \cdot \sin \frac{\pi}{4} = 1 \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2}
\]

Thus, the point on the curve is \( \left( \frac{\pi}{4}, \frac{\sqrt{2}}{2} \right) \).

### Step 4: Write the equation of the tangent line

Using the point-slope form \( y - y_1 = m(x - x_1) \) where \( m = f'(\frac{\pi}{4}) \) and \( (x_1, y_1) = \left( \frac{\pi}{4}, \frac{\sqrt{2}}{2} \right) \):
\[
y - \frac{\sqrt{2}}{2} = \frac{3\sqrt{2}}{2} \left( \theta - \frac{\pi}{4} \right)
\]

This is the equation of the tangent line at \( \theta = \frac{\pi}{4} \).

---

Would you like more details or have any questions about the solution?

### Related Questions
1. How would the process change if \( f(\theta) = \tan \theta + \sin \theta \) instead?
2. What is the slope of the tangent to \( f(\theta) = \cos \theta \tan \theta \) at \( \theta = \frac{\pi}{3} \)?
3. Can we find a similar tangent line if \( f(\theta) = \sin \theta \cdot \cos \theta \)?
4. How would we differentiate \( f(\theta) = \frac{\tan \theta}{\sin \theta} \) using the quotient rule?
5. What is the general method for finding tangent lines of trigonometric functions?

#### Tip:
When differentiating products like \( \tan \theta \sin \theta \), consider using product or quotient rules as appropriate to simplify

Find the equation of the line tangent to the graph of f(θ) = tanθ sinθ when θ = π/4. Use the quotient rule.

Learn how to find the tangent line to the function f(θ) = tanθ sinθ at θ = π/4 using the quotient rule. This solution covers the steps for differentiating trigonometric functions and calculating the tangent line equation, suitable for advanced high school or college 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