Trigonometry

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Using the Double-Angle Identity for Cosine

Identities for angles that are twice as large as one of the common angles (double angles) are used frequently in trig. These identities allow you to deal with a larger angle in the terms of a smaller and [more…]

How to Use Half-Angle Identities to Find the Sine of an Angle

By adding, subtracting, or doubling angle measures, you can find lots of exact values of trigonometry functions using the functions of angles you already know. For example, even though you can use a difference [more…]

How to Work Both Sides of a Trig Identity

With a trigonometry identity, working on both sides of the equation is even more fun than working on both sides of an algebraic equation. In algebra, you can multiply each side by the same number, square [more…]

Change to Sines and Cosines in a Trigonometry Identity

With some trig identities, you may decide to simplify matters by either changing everything to sines and cosines or by factoring out some function. Sometimes, it isn’t clear which side you should work [more…]

When to Factor a Trigonometry Identity

You’ll know that you need to factor a trig identity when powers of a particular function or repeats of that same function are in all the terms on one side of the identity. [more…]

Break Up or Combine Fractions to Solve a Trigonometry Identity

A trig identity with fractions can work to your advantage; you’re given a plan of attack. You can work toward getting rid of the fraction and, in the process, solve the problem. Two of the main techniques [more…]

How to Remove a Third Angle to Solve a Trigonometry Identity

Sum and difference identities usually involve two different angles and then a third combined angle. When proving these trig identities, you often need to get rid of that third angle. The following example [more…]

How to Find the Inverse of a Trig Function

You use inverse trigonometry functions to solve equations such as sin x = 1/2, sec x = –2, or tan 2x = 1. In typical algebra equations, you can solve for the value of [more…]

How to Distinguish between Trigonometry Functions and Relations

Technically, an inverse trig function is supposed to have only one output for each input. (Part of the definition of an inverse is that the function and inverse are one-to-one.) With any one-to-one function [more…]

Identify the Domains and Ranges of Inverse Trigonometry Functions

A function that has an inverse has exactly one output (belonging to the range) for every input (belonging to the domain), and vice versa. To keep inverse trig functions consistent with this definition, [more…]

The Trigonometry Functions Table

You can use this table of values for trig functions when solving problems, sketching graphs, or doing any number of computations involving trig. The values here are all rounded to three decimal places. [more…]

How to Find Solutions for a Multiple-Angle Trigonometry Function

Multiple-angle trig functions include [more…]

Rewrite a Simple Trigonometry Equation Using an Inverse to Solve It

The simplest type of trigonometry equation is the one that you can immediately rewrite as an inverse in order to determine the solutions. Some examples of these types of equations include: [more…]

How to Solve a Trigonometry Equation by Factoring Quadratics

Quadratic equations are nice to work with because, when they don’t factor, you can solve them by using the quadratic formula. The types of quadratic trig equations that you can factor are those like tan [more…]

How to Factor Trigonometry Expressions with Degrees Higher than 2

Although factoring quadratics is a breeze, factoring trigonometry equations with higher degrees can get a bit nasty if you don’t have a nice situation such as just two terms or a quadratic-like equation [more…]

Find the Area of a Triangle Using ASA

When you have two angles in a triangle and the side between them (ASA), you can use trig to find the area of the triangle. The formulas go as follows.

In triangle [more…]

Comparing Cosine and Sine Functions in a Graph

The relationship between the cosine and sine graphs is that the cosine is the same as the sine — only it’s shifted to the left by 90 degrees, or π/2. The trigonometry equation that represents this relationship [more…]

Graph the Asymptote of a Tangent Function

An asymptote is a line that helps give direction to a graph of a trigonometry function. This line isn’t part of the function’s graph; rather, it helps determine the shape of the curve by showing where [more…]

Graph the Asymptotes of a Cotangent Function

The graphs of the tangent function lay the groundwork for the graphs of the cotangent function. After all, the tangent and cotangent are cofunctions and reciprocals, and have all sorts of connections. [more…]

How to Find the Area of a Triangle with SAS

When you know the lengths of two of a triangle’s sides plus the measure of the angle between those sides (SAS), you can find the area of the triangle. This method requires a little trigonometry — you have [more…]

Find Opposite-Angle Trigonometry Identities

The opposite-angle identities change trigonometry functions of negative angles to functions of positive angles. Negative angles are great for describing a situation, but they aren’t really handy when it [more…]

Right Triangle Definitions for Trigonometry Functions

The basic trig functions can be defined with ratios created by dividing the lengths of the sides of a right triangle in a specific order. The label hypotenuse [more…]

Coordinate Definitions for Trigonometry Functions

The trig functions can be defined using the measures of the sides of a right triangle. But they also have very useful definitions using the coordinates of points on a graph. First, let let the vertex of [more…]

Signs of Trigonometry Functions in Quadrants

An angle is in standard position when its vertex is at the origin, its initial side is on the positive x-axis, and the terminal side rotates counterclockwise from the initial side. The position of the [more…]

Laws of Sines and Cosines

The laws of sines and cosines give you relationships between the lengths of the sides and the trig functions of the angles. These laws are used when you don’t have a right triangle — they work in any triangle [more…]

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