3.13 Trigonometry and Polar Coordinates

The (x,y) coordinate system is called the Cartesian coordinate system. However, this system of plotting coordinates is not the only system that exists. There are many coordinate systems, like the polar and parametric coordinate systems. In this guide, we will focus on the polar coordinate system.

The polar coordinate system is a method of representing points in two-dimensional space. Unlike the traditional Cartesian coordinate system, which uses a pair of perpendicular lines to define a point, the polar coordinate system uses a radial line from the origin to the point, and an angle from the positive x-axis.

In this system, each point is represented by a pair of values: the radial distance from the origin, known as the "magnitude" or "radius", and the angle. The magnitude is denoted by "r", while the angle is denoted by "θ".

The polar coordinate system is particularly useful in cases where it is easier to describe a point in terms of its distance from the origin and its angle, such as in cases involving circles or other circular shapes. It can also be used to solve complex problems in physics and engineering, where Cartesian coordinates may be less intuitive.

As shown in the image above, the polar plane is made up of circles centered at the origin. In the polar coordinate system, the "pole" is the origin, defined as the point (0,0). The origin serves as the center of the coordinate system, and all other points in the plane are represented by their distance and angle from the origin.

The "polar axis" is the positive x-axis, which is defined as the line where the angle is equal to 0 degrees. The polar axis serves as the reference line from which all other angles are measured. The polar axis can be thought of as the line pointing directly to the right from the origin, and all other angles are measured counterclockwise from this line. The angle can be measured in degrees or radians. Sound familiar? This is the same as how angles are measured in the unit circle. In the unit circle, the distance from the origin is 1 unit, but in the polar plane, the distance can vary as well.

Together, the pole and the polar axis form the backbone of the polar coordinate system. All points in the plane can be represented in terms of their radial distance from the origin and angle from the polar axis.

In the Cartesian coordinate system, one point only has one set of coordinates that represent that point. The point (3, 4) can only be represented by these coordinates, and no other set of coordinates can be used to reference this point.

In the polar coordinate system, however, a point in the plane can be represented by multiple sets of polar coordinates. This is because the radial distance from the origin, "r", and the angle "θ" are only defined up to a positive or negative sign, and up to a multiple of 2𝛑.

Consider the point P in the plane with polar coordinates (-2, 30°). The same point can also be represented by the following polar coordinates:

All of these polar coordinates represent the same point P in the plane, but with different radial distances and angles. For example, the point (2, 210°) represents the same point as (-2,30°), but with a negative radial distance, meaning the point is reflected across the origin. Similarly, the point (-2, 390°) represents the same point as (2,30°), but with an angle of 390°, which is equivalent to 30° + 360°. And finally, the point (2, -150°) represents the same point as (2,30°), but with a negative radial distance and an angle of negative 150°, which is equivalent to 30° - 180°.

The points (r,θ) and (-r,θ + 180°) represent the same point in the plane. This is because changing the sign of the radial distance results in a reflection of the point across the origin. Adding 180° cancels out the reflection to give you your original point.

Similarly, the point (r,θ) and (r,θ + 2𝛑) represent the same point in the plane. This is because adding multiples of 2𝛑 to the angle results in the same point on the unit circle, but with a different polar coordinate representation.

To convert from polar to Cartesian coordinates, we use the following formulas:

where r is the radial distance from the origin, θ is the angle with the polar axis, x is the x-coordinate, and y is the y-coordinate.

The conversion from polar to Cartesian coordinates involves converting the radial distance and angle into x and y coordinates using basic trigonometry. The cosine of the angle gives the x-coordinate, and the sine of the angle gives the y-coordinate.

These formulas are derived using trigonometry and right triangles. Drawing a right triangle from the origin to the point in question, r is the hypotenuse, and x and y are the side lengths. This is very similar to the unit circle, where the hypotenuse was simply 1.

To convert the polar coordinates (2, 60°) to Cartesian coordinates, we use the formulas:

x = rcos(θ) = 4 * cos(60°) = 4 * (√3/2) = 2√3 y = rsin(θ) = 4 * sin(60°) = 4 * (1/2) = 2

So the point P in the polar coordinate system is equivalent to the point (2√3, 2) in the Cartesian coordinate system.

To convert from Cartesian to polar coordinates, we use the following formulas:

where x is the x-coordinate, y is the y-coordinate, r is the radial distance from the origin, and θ is the angle with the polar axis.

The conversion from Cartesian to polar coordinates involves finding the radial distance and angle of a point in the plane using basic trigonometry. The radial distance is found using the Pythagorean theorem, and the angle is found using the inverse tangent function.

Consider the point (x, y) = (3, 4) in the Cartesian coordinate system. To convert these Cartesian coordinates to polar coordinates, we use the following formulas:

r = √(x^2 + y^2) = √(3^2 + 4^2) = 5 θ = tan^-1(y/x) = tan^-1(4/3) = 53.13°

So the point (x, y) = (3, 4) in the Cartesian coordinate system is equivalent to the point (r, θ) = (5, 53.13°) in the polar coordinate system. Be sure to check whether the angle is in degrees or radians so you can use the appropriate mode on your calculator for the arctangent function.

When using the arctangent function, we have to remember that it will only output a value between -𝛑/2 and 𝛑/2. However, there are more angles in the polar plane than just this range of angles, so we may have to manipulate our output from the arctangent function to arrive at the correct answer. The general rule is that if the x-coordinate of the point you are converting is above 0 (positive), then you can use the output of the arctangent function as is. However, if the x-coordinate is negative, you will need to add 𝛑 or 180° to your answer to convert to the correct point. This is because the arctangent function only outputs angles in the first and fourth quadrant of the coordinate plane (the right side), but if the x-coordinate is negative, then the angle is in the second or third quadrant (the left side). Therefore, we need to add 𝛑 to our answer to ensure we arrive at the correct point.

Complex numbers are an extension of real numbers that allow us to perform arithmetic with numbers that have a real and imaginary component. In the Cartesian plane, complex numbers are represented as points, with the real component as the x-coordinate and the imaginary component as the y-coordinate.

A complex number is written in the form a + bi, where a is the real component and bi is the imaginary component. The imaginary unit i is defined as the square root of -1. Remember that the square root of a negative number isn't a real number, so we call it imaginary.

For example, the complex number 3 + 4i can be represented as the point (3, 4) in the Cartesian plane. We can perform arithmetic operations with complex numbers, such as addition, subtraction, multiplication, and division, in a way that is similar to real numbers.

Complex numbers can be graphed in the complex plane, which is a two-dimensional coordinate system where the x-axis represents the real component and the y-axis represents the imaginary component. This graphical representation allows us to visualize complex numbers in a way that is similar to real numbers in the Cartesian plane.

The polar representation of a complex number is given by:

where r is the magnitude of the complex number and θ is the argument, measured in radians or degrees.

The magnitude r represents the length of the ray representing the complex number in the complex plane, while the argument θ represents the direction of the ray. The real part of the complex number is given by r * cos(θ), while the imaginary part is given by i(r * sin(θ)).

The x and y representations can be seen as analogous to the magnitude r and argument θ in the polar representation. Just as r determines the length of the ray representing the complex number in the polar plane, x and y determine the length of the ray in the Cartesian plane. And just as θ determines the direction of the ray in the polar plane, the x and y coordinates determine the direction of the ray in the Cartesian plane.