Find all points of intersection (r,θ) of the Lemniscate r^2=36cos(2θ) , and the circle r=4 for values of θ satisfying −π/2<θ<π/2 .



We are tasked with finding the points of intersection between the **Lemniscate** given by:

\[
r^2 = 36 \cos(2\theta)
\]

and the **circle** given by:

\[
r = 4
\]

### Step 1: Substitute \( r = 4 \) into the Lemniscate equation

Substitute \( r = 4 \) into the Lemniscate equation \( r^2 = 36 \cos(2\theta) \):

\[
4^2 = 36 \cos(2\theta)
\]

Simplifying:

\[
16 = 36 \cos(2\theta)
\]

Solve for \( \cos(2\theta) \):

\[
\cos(2\theta) = \frac{16}{36} = \frac{4}{9}
\]

### Step 2: Solve for \( \theta \)

Now, solve for \( 2\theta \):

\[
2\theta = \cos^{-1}\left(\frac{4}{9}\right)
\]

Using the inverse cosine function:

\[
2\theta = \pm \cos^{-1}\left(\frac{4}{9}\right)
\]

Thus, we have:

\[
\theta = \pm \frac{1}{2} \cos^{-1}\left(\frac{4}{9}\right)
\]

### Step 3: Calculate \( \cos^{-1}\left(\frac{4}{9}\right) \)

Let's compute the value of \( \cos^{-1}\left(\frac{4}{9}\right) \):

\[
\cos^{-1}\left(\frac{4}{9}\right) \approx 1.12 \text{ radians}
\]

Now, solve for \( \theta \):

\[
\theta = \pm \frac{1.12}{2} \approx \pm 0.56 \text{ radians}
\]

### Step 4: Determine points of intersection

The values of \( \theta \) are \( \theta \approx 0.56 \) and \( \theta \approx -0.56 \).

### Step 5: Find corresponding values of \( r \)

For both values of \( \theta \), the value of \( r = 4 \) since the equation of the circle is \( r = 4 \).

Thus, the points of intersection are:

\[
(r, \theta) = (4, 0.56) \quad \text{and} \quad (4, -0.56)
\]

### Final Answer:

The points of intersection are:

\[
(4, 0.56) \quad \text{and} \quad (4, -0.56)
\]

---

Would you like a detailed explanation on any step? 

Here are 5 related questions you can explore:
1. How is the polar form of a Lemniscate derived?
2. What is the geometric interpretation of \( \cos(2\theta) \) in polar coordinates?
3. How can we solve \( \cos^{-1} \) expressions algebraically without a calculator?
4. What are the properties of a Lemniscate?
5. How can we find the area enclosed by a Lemniscate?

**Tip:** Always remember to check the interval of \( \theta \) when solving polar equations!

Find all points of intersection (r,θ) of the Lemniscate r^2 = 36cos(2θ), and the circle r = 4 for values of θ satisfying −π/2 < θ < π/2.

Explore how to find the points of intersection between the Lemniscate r^2 = 36cos(2θ) and the circle r = 4 for values of θ in the range −π/2 < θ < π/2. This problem involves key polar coordinate and trigonometric concepts, with a step-by-step solution.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation