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Trigonometry

Pythagorean Theorem

The Pythagorean Theorem states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides (legs).

It doesn't matter which leg you assign as a and which leg you assign as b.

Example

Find the hypotenuse of the trianlge below.

Special Right Triangles

There are two main types of special right triangles. Special right triangles are right triangles that, given specific angle measures, always have the same side ratios. These allow you to solve for the sides without using the Pythagorean Theorem.

Trigonometric Functions

Before, we used Pythagoras' Theorem when we knew the length of two sides. However, if we are given a problem with only one side length and the measure of an angle other than a right angle, we can use trigonometric functions. Sine, Cosine and Tangent are the three main functions in trigonometry, which are abbreviated as sin, cos and tan respectively. Other trigonometric functions include cosecant, secant, cotangent, which are the reciprocals of the three main functions, and are abbreviated as csc, sec, and cot respectively. The calculation of each trigonometric function is one side of a right triangle divided by another side. To determine which sides of the triangles to divide, use the acronym SOH-CAH-TOA, pronounced “So Cah Toe-ah.”

Example

As long as we are given an angle and one side of the right triangle, we can use proportions to solve for the missing side(s).

To determine what ratio to use, we must see what is given and what we have to solve for, relative to the given angle. Side a is opposite the given angle, and the hypotenuse, 7, is given, so we will use the ratio that involves the opposite and hypotenuse. The trigonometric identity that combines these is sine.

Reciprocal Trig Functions

Each of the three main trig functions have a reciprocal function. A reciprocal is like an upside down fraction.

The trigonometric ratios of the reciprocal trig functions are just like that of their corresponding main function, but flipped.

Inverse Trig Functions

While normal trig functions use an angle measure to find a ratio, inverse trig functions use a ratio to find an angle.

Inverse Trig Functions are NOT the same as reciprocal trig functions. The reciprocal trig functions also have inverses.

Trig Identities

Trigonometric identities are equations that remain true regardless of the given angle measure. Below are some common trig identities that are important to know.

Unit Circle

The Unit Circle is a circle with a radius of 1 that can easily illustrate the relationship between different angles and their respective sines and cosines. Angles on the unit circle can be measured in either radians or degrees.