Convert the polar equation to a rectangular one. -7r=tan(theta)sec(theta)

To convert the polar equation \(-7r = \tan(\theta) \sec(\theta)\) into rectangular form, we need to use the relationships between polar and rectangular coordinates:

1. \( r = \sqrt{x^2 + y^2} \)
2. \( x = r \cos(\theta) \)
3. \( y = r \sin(\theta) \)
4. \(\tan(\theta) = \frac{y}{x}\)
5. \(\sec(\theta) = \frac{r}{x}\)

Let's rewrite the equation step-by-step:

### Step 1: Substitute for \(\tan(\theta)\) and \(\sec(\theta)\)
The original equation is:
\[
-7r = \tan(\theta) \sec(\theta)
\]
Using \(\tan(\theta) = \frac{y}{x}\) and \(\sec(\theta) = \frac{r}{x}\), we can rewrite \(\tan(\theta) \sec(\theta)\) as:
\[
\tan(\theta) \sec(\theta) = \frac{y}{x} \cdot \frac{r}{x} = \frac{yr}{x^2}
\]
So the equation becomes:
\[
-7r = \frac{yr}{x^2}
\]

### Step 2: Divide both sides by \(r\) (assuming \(r \neq 0\))
\[
-7 = \frac{y}{x^2}
\]

### Step 3: Multiply both sides by \(x^2\) to eliminate the fraction
\[
-7x^2 = y
\]

### Final Answer
The rectangular form of the equation is:
\[
y = -7x^2
\]

This is the rectangular equation equivalent to the given polar equation.

Would you like more details on each step or have any questions? 

---

Here are some related questions:
1. How do we convert polar equations to rectangular form in general?
2. What are the common polar-to-rectangular coordinate transformations?
3. What role do trigonometric identities play in these conversions?
4. How would we convert \(-7r = \cos(\theta) \sin(\theta)\) to rectangular form?
5. Can we convert rectangular equations back into polar form?

**Tip:** Always double-check trigonometric identities when converting between coordinate systems, as they can simplify the process significantly.

Discover how to convert the polar equation -7r = tan(θ) sec(θ) into rectangular form with detailed steps and essential trigonometric identities. Perfect for high school students learning about coordinate transformations and trigonometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation