2r = a sin 2 at 4   = find the the per from tangent

The equation you've given is \( 2r = a \sin(2\theta) \), and you're asking about the tangent and possibly its properties at \( \theta = 4\pi \).

Let's break this down step-by-step to interpret it:

1. **Given Equation:**
   \[
   2r = a \sin(2\theta)
   \]
   Simplifying, we get:
   \[
   r = \frac{a}{2} \sin(2\theta)
   \]

2. **At \( \theta = 4\pi \):**
   \[
   r = \frac{a}{2} \sin(8\pi)
   \]
   Since \( \sin(8\pi) = 0 \) (as the sine function repeats every \( 2\pi \)), we have:
   \[
   r = 0
   \]
   At \( \theta = 4\pi \), the radius \( r \) is zero, which means the point lies at the origin.

3. **Finding the Tangent:**
   Since \( r = 0 \) at \( \theta = 4\pi \), we are dealing with a point at the origin. The tangent to a polar curve at the origin is often undefined because the curve passes through the origin and doesn't have a clear direction at that point.

   However, if you wanted to compute the tangent to this curve at other points, you would typically differentiate the polar equation \( r = \frac{a}{2} \sin(2\theta) \) with respect to \( \theta \) and use the result to find the slope of the tangent line at a given angle \( \theta \).

---

Would you like more detailed steps on finding the tangent, or have any questions?

### Related Questions:
1. What is the general formula for the slope of a tangent line in polar coordinates?
2. How do we calculate the derivative of \( r = \frac{a}{2} \sin(2\theta) \) with respect to \( \theta \)?
3. How does the sine function's periodicity affect polar curves?
4. How do you find the points where a polar curve intersects the origin?
5. What is the geometric significance of tangents at the origin in polar coordinates?

### Tip:
In polar coordinates, the tangent line at any point can be tricky, especially near the origin, so always carefully differentiate the equation.

2r = a sin 2θ at 4π θ = find the per from tangent

Explore how to find the tangent line for the polar curve defined by the equation 2r = a sin(2θ) at θ = 4π. Understand the step-by-step process, including differentiation in polar coordinates, and the periodic properties of trigonometric functions. Suitable for high school students in Grades 11-12 studying advanced mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation