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### The Cosecant Function

The *cosecant* function, abbreviated *csc*, is called a reciprocal function because it is the reciprocal of one of the three basic trig functions, sine. The cosecant function uses the ratio [more…]

### The Cotangent Function

The *cotangent* function, abbreviated *cot*, is called a reciprocal function because it uses the reciprocal of the trigonometry function, tangent (hence, the [more…]

### How to Place Points and Fractions on a Unit Circle

To work with trigonometry functions, it’s helpful to be able to place points and fractions on a unit circle. The unit circle is a circle with its center at the origin of the coordinate plane and with a [more…]

### Negative and Positive Angles Cutting a Circle

The circle with its center at the origin is a platform for describing all the possible angle measures from 0 to 360 degrees, plus all the negatives of those angles, and plus all the multiples of the positive [more…]

### How to Locate Reference Angles

Each of the angles positioned with its center at the origin has a *reference* angle, which is always a positive acute angle. By identifying the reference angle, you can determine the function values for [more…]

### A Quick Table for the Three Reciprocal Trig Functions

Reciprocal functions have values that are reciprocals, or flips, of the values for their respective functions. The reciprocal of sine is cosecant, so each function value is the reciprocal of the corresponding [more…]

### How to Circumscribe a Triangle

Every triangle can be circumscribed by a circle, meaning that one circle — and only one — goes through all three vertices (corners) of any triangle. In laymen’s terms, any triangle can fit into some circle [more…]

### How to Divide a Line Segment into Multiple Parts

If you can find the midpoint of a segment, then you can divide it into two equal parts. Finding the middle of each of the two equal parts allows you to find the points needed to divide the entire segment [more…]

### How to Locate the Center of a Circle

You can use algebra to find the center of a circle. If the endpoints of one diameter of a circle are (*x*_{1},*y*_{1}) and (*x*_{2},*y*_{2}), then the center of the circle has the following coordinates: [more…]

### Using the 30-60-90 Right Triangle

A 30-60-90 right triangle has angles measuring just what the name says. The two acute, complementary angles are 30 and 60 degrees. These triangles are great to work with, because the angle measures, all [more…]

### The Secant Function

The *secant* function, abbreviated *sec*, is called a reciprocal function because it uses the reciprocal of the trig function, cosine. The secant function uses the ratio [more…]

### How to Compute Reference Angles in Degrees

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

### Tangent and Secant Identities on a Unit Circle

Starting with the Pythagorean identity, sin^{2}θ + cos^{2}θ = 1, you can derive tangent and secant Pythagorean identities. All you do is throw in a little algebra and apply the reciprocal and ratio identities [more…]

### How to Use the Angle-Sum Identity When You Don’t Know the Angle

In some trigonometry problems, you may not know what the measure of an angle is, but you know something about the angle’s function values. For example, suppose you have two angles, α in the second quadrant [more…]

### How to Use the Subtraction Identities in a Trig Problem

You can find function values of angles using angle-addition identities. And you have more possibilities for finding the function values of angles when you use subtraction in a trig problem. For example [more…]

### Find Trigonometry Ratio Identities

Trig has two identities called *ratio identities*. This label can be confusing, because all the trig functions are defined by ratios. Somewhere along the line, however, mathematicians thought this description [more…]

### Assign Negative and Positive Trig Function Values by Quadrant

The first step to finding the trig function value of one of the angles that’s a multiple of 30 or 45 degrees is to find the reference angle in the unit circle. When the reference angle comes out to be [more…]

### Domain and Range of Sine and Cosine Functions

The *domain* of a sine or cosine trigonometry function consists of all the input values that the function can handle — the way the function is defined. Of course, you want to get output values [more…]

### Domain and Range of Cosecant and Secant Trigonometry Functions

The *domain* of a cosecant or secant trig function consists of all the input values that the function can handle — the way the function is defined. Of course, you want to get output values [more…]

### Domain and Range of Tangent and Cotangent Trigonometry Functions

The *domain* of a tangent or cotangent trig function consists of all the input values that the function can handle — the way the function is defined. Of course, you want to get output values [more…]

### Basic Pythagorean Identities for Trigonometry Functions

The Pythagorean identities are building blocks for many of the manipulations of trigonometric equations and expressions. They provide a greater number of methods for solving trig problems more efficiently [more…]

### Express Sine in Terms of Secant or Cosecant

Even though each trig 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 may have [more…]

### How to Determine the Altitude of a Balloon

Trigonometry has many applications for finding distances. For example, to find the altitude of a floating object, you can use the angle between one sighting of the object and a second sighting to solve [more…]

### The Origin of the Half-Angle Identities for Sine

The trig identities come in sums, differences, ratios, multiples, and halves. With a half-angle identity, you can get the value of a sine for a 15-degree angle using a function of of 30 degree angle. You [more…]

### How to Find Half-Angle Identities for Tangent

The half-angle trig identity for tangent has two versions. Rather than this being a nuisance, having more than one option is really rather nice, because you can choose the version that works best for your [more…]