# Polar to Rectangular Conversion

• Dooh
In summary, The person is struggling with converting polar equations into rectangular form and is seeking help. They suggest using the triple angle formula to reduce the equation, but the person only knows the double angle formula. They are then guided to use the sum of angles for the sine function and the correct formula is provided. The person then asks if they can factor out the sine and cosine terms, but is corrected and given the correct formula.
Dooh
Can someone help me with these problems? It's been bugging me i can't seem to solve it.

Lets assume T = theta

I can't seem to find a way to convert these polar equations into rectangular form.

r = 2 sin 3T

r = 6 / 2 - 3 sinT

If possible, can someone help me with this and list it in a step-by-step fashion so i can see how you get the answer. Thanks.

Last edited:
$$r = 2 \sin (3 \theta)$$

$$r^2 = 2r \sin (3 \theta)$$

then try the triple angle formulas to reduce $\sin (3 \theta)$

Sorry but the furthest that our teacher had taught us in the double angle formula.

It's not hard. Why don't you try it?

$$\sin (3 \theta) = \sin (\theta + 2 \theta)$$

Use the sum of angles for the sin

$$\sin (\alpha + \beta) = \sin \alpha cos \beta + \cos \alpha \sin \beta$$

Ok, so i got:

$$r^2 = 2r (\sin \theta \cos 2 \theta + \cos \theta \sin 2 \theta )$$

can i take out the $$\sin \theta \cos \theta$$

and get

$$r^2 = 2r \sin \theta \cos \theta (\cos \theta + \sin \theta )$$

or should i further expand the equation?

No that's wrong it will be

$$\sin (3 \theta) = 3 \sin \theta - 4 \sin^{3} \theta$$

thus

$$r^2 = 2r (3 \sin \theta - 4 \sin^{3} \theta)$$

## 1. What is the polar to rectangular conversion?

The polar to rectangular conversion is a mathematical process used to convert coordinates from the polar coordinate system to the rectangular coordinate system. This allows for easier representation and calculation of points on a graph or in a two-dimensional plane.

## 2. How is the polar to rectangular conversion performed?

The polar to rectangular conversion involves using trigonometric functions to determine the x and y coordinates of a point in the rectangular coordinate system. The formula is x = r * cos(θ) and y = r * sin(θ), where r is the distance from the origin to the point and θ is the angle formed between the positive x-axis and the line connecting the origin to the point.

## 3. Why is the polar to rectangular conversion useful?

The polar to rectangular conversion is useful because it allows for easier visualization and calculation of points in a two-dimensional plane. It also allows for the representation of complex numbers using the real and imaginary axes, making it useful in fields such as engineering, physics, and mathematics.

## 4. What are the differences between polar and rectangular coordinates?

Polar coordinates use distance and angle to represent a point, while rectangular coordinates use x and y coordinates. Polar coordinates are useful for representing circular or rotational motion, while rectangular coordinates are better for representing linear motion. Additionally, polar coordinates have an infinite number of representations for a single point, while rectangular coordinates have a unique representation for each point.

## 5. Can polar to rectangular conversion be applied to three-dimensional coordinates?

Yes, polar to rectangular conversion can be applied to three-dimensional coordinates by adding a z-coordinate to the equation. The formula becomes x = r * cos(θ) * cos(φ), y = r * sin(θ) * cos(φ), and z = r * sin(φ), where r is the distance from the origin to the point, θ is the angle formed between the positive x-axis and the line connecting the origin to the point in the xy-plane, and φ is the angle formed between the positive z-axis and the line connecting the origin to the point in the xz-plane.

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