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### Commonly Used Values of Selected Trig Functions

When performing transformations in trig functions, such as rotations, you need to use the numerical values of these functions. Here are some of the more commonly used angles. [more…]

### Classifying Differential Equations by Order

The most common classification of differential equations is based on order. The order of a differential equation simply is the order of its highest derivative. You can have first-, second-, and higher-order [more…]

### Distinguishing among Linear, Separable, and Exact Differential Equations

You can distinguish among linear, separable, and exact differential equations if you know what to look for. Keep in mind that you may need to reshuffle an equation to identify it. [more…]

### Defining Homogeneous and Nonhomogeneous Differential Equations

In order to identify a nonhomogeneous differential equation, you first need to know what a homogeneous differential equation looks like. You also often need to solve one before you can solve the other. [more…]

### Using the Method of Undetermined Coefficients

If you need to find particular solutions to nonhomogeneous differential equations, then you can start with the method of undetermined coefficients. Suppose you face the following nonhomogeneous differential [more…]

### 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 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 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…]

### Pythagorean Sine and Cosine Identities on a Unit Circle

If you have ever wondered why the Pythagorean identity, sin^{2}θ + cos^{2}θ = 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, sin^{2}*θ* + cos^{2}*θ* = 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…]

### 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…]

### 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…]

### 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…]

### 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…]

### 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…]

### 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…]