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What Is the Pythagorean Theorem and When Is It Used?

What is the Pythagorean theorem: Graph of how the sum of the squares creates the right triangle ABC

What is the Pythagorean theorem? It’s a trigonometry equation used to find the length of one side of a right triangle. Though similar concepts had been discovered by the Babylonians, Greek Mathematician Pythagoras was the first person to come up with a geometric proof about how the sum of the squares of the lengths can determine the side lengths of a right triangle.

Pythagoras determined that when three squares are arranged so that they form a right angle triangle, the largest of the three squares must have the same area as the other two squares combined. In the picture below, you can see how the sum of the squares creates the right triangle ABC.

This realization about the area of the squares led to the Pythagoras theorem:

What is the Pythagorean theorem: a2+b2=c2 formula

Squares are different from other parallelograms and trapezoids because all their sides are equal lengths. So since squares are made up of four equal sides, you can see that each individual square makes up a side of the right triangle.

The length of the largest square, which we'll call length c, is the length of the hypotenuse. (The hypotenuse is the longest side of a right triangle.) The smaller squares make up the other two sides of the right triangle.

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Here’s How Vector Subtraction Works

Graph showing the process of vector subtraction

A vector is a term used to define any line segments with a specified starting and ending point. All vectors are drawn at an angle and have a specified direction. Learning to subtract vectors is helpful when you need to see how much one vector must travel to get to the other.

Vector subtraction is the process of subtracting the coordinates of one vector from the coordinates of a second vector.

See the example below. The coordinates of vector a are marked as (3,3) and the coordinates of vector b as (1, 2).

When subtracting vectors, you must take the first vector quantities and subtract the second quantity. Let's subtract vector b from vector a:

Formula for vector subtraction

Formula for vector subtraction

Formula for vector subtraction

Your resultant vector coordinates for this particular example are (2, 1).

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Sine Function, Cosine Function, and Tangent Function Explained

Sine function in a right triangle

There are three basic functions to understand in trigonometry: sine function, cosine function, and tangent function. Each function takes in an angle measure, which is called theta and represented by the symbol sine function: symbol θ. The angle measure is usually expressed in radians or degrees. The output of the function is a ratio comparing one side of the triangle to another.

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How To Find the Length of a Triangle

how to find the length of a triangle: Example of the Law of Sines in action

You’ll be asked how to find the length of a triangle over and over again in math and trigonometry. Maybe you need to find the missing side of a right triangle, maybe you know both side b and side c, or maybe you know only the opposite angle of the length of a side you are trying to find. In any case, we have formulas to help.

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How To Find the Polar Form of a Complex Number

Graph showing a complex number graphed in rectangular coordinates

Polar form is a way of representing complex numbers by graphing them. But before determining how to find the polar form, we must first establish what a complex number is. A complex number is a trigonometry concept that means a “combination of real numbers and imaginary numbers.” Real numbers can be positive or negative. They are essentially any quantifiable number that, when squared, results in a positive value.

Imaginary numbers are numbers that don't compute on a calculator. When squared, they result in a negative value. Because these numbers are imaginary, they cannot be expressed with numerical values. Instead, the unit used for imaginary numbers is i, the square root of -1. Here are some examples of imaginary numbers: 5i, 3.6i, −14.3i, (√7)i, 2,891i. All of these numbers have been multiplied by the value of i or √−1.

Complex numbers have a real part and an imaginary part. Complex number z is used to represent the combined result of the real and imaginary parts. Here are some examples of complex numbers:

polar form: Example of a complex number

Example of a complex number

Example of a complex number

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