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Trigonometry

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How to Use the Double-Angle Identity for Sine

The double-angle formula for sine comes from using the trig identity for the sine of a sum, sin (α + β) = sinαcosβ + cosαsinβ. If α = β, then you can replace β with α in the formula, giving you [more…]

How to Change Degrees to Radians

Many math problems require you to change measurements from degrees to radians. Degree measures are more familiar than radians to most people. A circle is divided into 360 [more…]

How to Use Radians to Solve a Trig Problem

Using radians is very helpful when you are doing trigonometry applications involving the length of an arc of a circle, which is part of its circumference. This might include measuring the sweep of a hand [more…]

How to Compute the Reference Angles in Radians

If you need to compute the measure (in radians) of the reference angle for any given angle θ, you can use the rules in the following table. [more…]

Use Coordinates of Points to Find Values of Trigonometry Functions

One way to find the values of the trig functions for angles is to use the coordinates of points on a circle that has its center at the origin. Letting the positive [more…]

How to Calculate Trigonometry Functions of Angles Using the Unit Circle

Calculating trig functions on angles using the unit circle is easy as pie. The following figure shows a unit circle, which has the equation x2 + y2 = 1, along with some points on the circle and their coordinates [more…]

How to Calculate Trigonometry Functions Using Any Circle

When determining the trig function values of angles graphed in standard position on a circle whose center is at the origin, you don’t have to have a unit circle to calculate coordinates. You can use a [more…]

How to Use Trigonometry to Find the Area of a Triangle

Sometimes, it’s hard to find the area of a triangle, especially when it’s not convenient to measure its height. When this happens, you can use trig to modify this traditional formula for the area of a [more…]

Pythagorean Sine and Cosine Identities on a Unit Circle

If you have ever wondered why the Pythagorean identity, sin2θ + cos2θ = 1, is so important, and where it came from, then read on. This identity is important because it sets an expression involving trig [more…]

Cotangent and Cosecant Identities on a Unit Circle

Starting with the Pythagorean identity, sin2θ + cos2θ = 1, you can derive cotangent and cosecant Pythagorean identities. All you do is throw in a little algebra and apply the reciprocal and ratio identities [more…]

Rearrange the Pythagorean Identities

Familiarizing yourself with the different versions of Pythagorean identities is helpful so that you can easily recognize them when solving trigonometry equations or simplifying expressions. [more…]

Express Sine in Terms of Cosine

Even though each trigonometry function is perfectly wonderful, being able to express each trig function in terms of one of the other five trig functions is frequently to your advantage. For example, you [more…]

Express Sine in Terms of Tangent

Even though each trigonometry function is perfectly wonderful, being able to express each trig function in terms of one of the other five trig functions is frequently to your advantage. For example, you [more…]

Express Sine in Terms of Cotangent

Even though each trigonometry function is perfectly wonderful, being able to express each trig function in terms of one of the other five trig functions is frequently to your advantage. For example, you [more…]

Labeling the Sides of a Right Triangle

A right triangle has two shorter sides, or legs, and the longest side, opposite the right angle, which is always called the hypotenuse. The two shorter sides have some other special names, too, based on [more…]

Determining if a Long Object Will Fit around a Hallway Corner

Here’s an application of trigonometry that you may be able to relate to: Have you ever tried to get a large piece of furniture around a corner in a house? You twist and turn and put it up on end, but to [more…]

Reciprocal Trigonometry Identities

The simplest and most basic trig identities (equations of equivalence) are those involving the reciprocals of the trigonometry functions. To jog your memory, a reciprocal of a number is 1 divided by that [more…]

How to Measure the Slope of a Road

Trigonometry functions have everyday applications. For example, you can use a trigonometry function to measure the slope of the surface you are standing on. [more…]

Angle of Elevation and Angle of Depression in Trigonometry Functions

Mathematical problems that require the use of trig functions often refer to one of two special angles: the angle of elevation or the angle of depression. The scenarios that use these angles usually involve [more…]

Using Trigonometry to See if a Ladder Reaches a Window

Every day, people use trigonometry to measure things that they can’t reach. How high is that building? Will this ladder reach to the top of that tree? By using the appropriate trig functions, you can find [more…]

How to Determine the Height of a Tree

You can use a trigonometry function to find the height of a tree while standing on the ground. For example, suppose you’re flying a kite, and it gets caught at the top of a tree. You’ve let out all 100 [more…]

How to Measure the Distance between Two Rooftops

Trigonometry functions have everyday applications. You can use a trig function to measure the distance between two rooftops — you just have to be standing on one of them. [more…]

How to Create a Table of Trigonometry Functions

In case your scientific calculator decides to conk out on you, you can construct a table that shows the exact values of trigonometry functions for the most commonly used angles. A bonus to this situation [more…]

The Tangent Function: Opposite over Adjacent

The trig function, tangent, is abbreviated tan. This function uses just the measures of the two legs and doesn’t use the hypotenuse at all. The tangent is described with this ratio: [more…]

Use One Function to Solve for Another Function

Sometimes you have to solve for a trigonometry function in terms of another function. For example, you may be given the cosine of an angle, and from that, you need to determine the sine and tangent of [more…]

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