Problem 67 - Algebra and Trigonometry

Question

Convert the polar equation to rectangular coordinates. $$ \sec \theta=2 $$

Step-by-Step Solution

Answer

The rectangular equation is \( y^2 = 3x^2 \).

Step 1: Understanding the Problem

We need to convert the polar equation \( \sec \theta = 2 \) into rectangular coordinates. Polar coordinates are represented as \( (r, \theta) \) while rectangular coordinates use \( (x, y) \). The key transformation relationships are \( x = r\cos\theta \), \( y = r\sin\theta \), and \( r = \sqrt{x^2 + y^2} \).

Step 2: Rewriting the Secant Function

The equation \( \sec \theta = 2 \) can be rewritten using the secant definition: \( \sec \theta = \frac{1}{\cos \theta} \). Thus, our equation becomes \( \frac{1}{\cos \theta} = 2 \).

Step 3: Solving for \(\cos \theta\)

To solve for \( \cos \theta \), take the reciprocal of both sides: \( \cos \theta = \frac{1}{2} \).

Step 4: Using Rectangular Conversion Formula

Substitute \( \cos \theta \) in the formula for \( x = r\cos\theta \): \( r\cos\theta = x \). Since \( \cos \theta = \frac{1}{2} \), this becomes: \( x = \frac{r}{2} \).

Step 5: Expressing \(r\) in Terms of \(x\)

We can solve the equation \( x = \frac{r}{2} \) to express \( r \) in terms of \( x \). This gives \( r = 2x \).

Step 6: Converting to Rectangular Equation

Using the relationship \( r^2 = x^2 + y^2 \), substitute \( r \) with \( 2x \): \( (2x)^2 = x^2 + y^2 \).

Step 7: Simplifying the Equation

Simplify \( (2x)^2 = x^2 + y^2 \) to get \( 4x^2 = x^2 + y^2 \). Subtract \( x^2 \) from both sides to solve for \( y^2 \): \( 3x^2 = y^2 \).

Key Concepts

Rectangular Coordinates

Rectangular coordinates are a way of representing points on a plane using two values: the horizontal distance (x-coordinate) and the vertical distance (y-coordinate) from the origin (0,0). This system is also known as Cartesian coordinates. Imagine a grid where each point can be pinpointed by moving along the x-axis (left or right) and the y-axis (up or down).

• The x-coordinate tells you how far to move horizontally from the origin.
• The y-coordinate tells you how far to move vertically from the origin.

Together, these coordinates make up the point (x, y) which precisely describes any location on the plane.
Using rectangular coordinates, you can perform a variety of mathematical operations including plotting points, determining distances between points, and performing algebraic calculations.

Polar Coordinates

Polar coordinates represent points in the plane using a different method than rectangular coordinates. Instead of using horizontal and vertical distances, polar coordinates use a distance from a central point (the origin) and an angle. This is helpful for problems involving circles and rotational motion.

• The distance from the origin is called the radius, denoted as \( r \).
• The angle, \( \theta \), is measured from the positive x-axis.

A point in polar coordinates is given as \((r, \theta)\). To convert from polar to rectangular coordinates, you can use the formulas:
- \( x = r \cos \theta \)- \( y = r \sin \theta \)
These formulas allow you to transform the polar coordinates into a form that can be used in a rectangular coordinate system.

Trigonometric Identities

Trigonometric identities are mathematical equations that relate the angles and sides of triangles. These identities are essential for converting between polar and rectangular coordinates. A few important ones include:

• The secant identity, \( \sec \theta = \frac{1}{\cos \theta} \), which is used in our problem to rewrite \( \sec \theta = 2 \) as \( \cos \theta = \frac{1}{2} \).
• Cosine and sine relationships, which connect polar and rectangular coordinates: \( x = r \cos \theta \) and \( y = r \sin \theta \).
• Pythagorean identity, \( r^2 = x^2 + y^2 \), helps in finding relationships between angles and sides.

These identities allow us to transform and understand equations in different coordinate systems, making them a powerful tool in trigonometry.