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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 Recognize Basic Trig Graphs

The graphs of the trig functions have many similarities and many differences. The graphs of the sine and cosine look very much alike, as do the tangent and cotangent, and then the secant and cosecant. [more…]

### How to Find the Midpoint of a Line Segment

To find the midpoint of a line segment, you just calculate the averages of the coordinates — easy as pie. The midpoint, *M*, of a segment with endpoints [more…]

### How to Pinpoint the Center of a Triangle

To find the center of a triangle, all you need are the locations of the three corners and the midpoints of the sides opposite those vertices. (It’s also a good idea to draw the triangle to help you see [more…]

### How to Recognize Parallel and Perpendicular Lines

There’s an easy way to tell whether two lines are parallel or perpendicular to each other in a graph — if you can determine the coordinates of some points on the lines: the lines are [more…]

### How to Use Function Notation

Function notation is like shorthand writing: if you understand it, you’ll save a lot of time (and pencil lead) when working out a problem. Without notation, defining a function or explaining how it works [more…]

### Determine Domain and Range in a Trig Function

When using functions, you can think of a domain as all the possible values you can put in for the input variable, and a range as all the possible values that you get back after performing the operations [more…]

### How to Find an Inverse Trig Function

Not all functions have inverses, and not all inverses are easy to determine. Here are some useful methods for finding inverses of basic algebraic functions. [more…]

### The Most Frequently Used Trig Angles

There are certain angles that you will use frequently when you work with trigonometry functions. You can find these angles by cutting a graph into four parts, and then dividing those parts into smaller [more…]

### Graph Angles in a Standard Position

In trigonometry and most other math topics, you draw angles in a standard, universal position, so that mathematicians around the world are drawing and talking about the same thing. [more…]

### Identify Coterminal Angles

An angle in *standard* *position* on the coordinate plane has its vertex at the origin and its *initial* (beginning) side along the positive *x*-axis. An angle’s [more…]

### How to Rename Coterminal Angles

Any angle can have many, many descriptions in terms of angle measures, because an angle measure is equivalent to the measures of its coterminal angles. The most frequently used positive angle measures [more…]

### How to Measure the Distance to the Moon Using Trig

One of the earliest applications of trigonometry was in measuring distances that you couldn’t reach, such as distances to planets or the moon or to places on the other side of the world. Here’s an example [more…]

### How to Measure the Speed of a Car around a Race Track

One of the great things about trigonometry is that you can use it to measure faraway things — or things that you don’t want to get too close to, like a race track. [more…]

### Define a Right Triangle and Its Parts

If you’re looking at their angles, triangles can be right, acute, or obtuse. So what makes a triangle *right*? Quite simply, a *right triangle* has a right angle in it. But it can only have one right angle [more…]

### How to Solve for a Missing Right Triangle Length

One of the nice qualities of right triangles is that you can use trigonometry to find the length of one side if you know the lengths of the other two sides. You don’t have this luxury with just any triangle [more…]

### How to Find the Distance across a Pond

Trigonometry is very handy for finding distances that you can’t reach to measure. Imagine that you want to string a cable diagonally across a pond (so you can attach a bunch of fishing line and hooks). [more…]

### Using Isosceles Right Triangles

The isosceles right triangle, or the 45-45-90 right triangle, has some special properties. For example, the two acute angles are equal, making the lengths of the two legs opposite them equal, too. What’s [more…]

### Use Six Different Ratios of a Right Triangle

Each of the three sides of a right triangle — hypotenuse, opposite, and adjacent — has a respective length or measure. And those three lengths or measures form six different ratios. Check out the following [more…]

### Identify Common Pythagorean Triples

A *Pythagorean* *triple* is a list of three numbers that fits the Pythagorean theorem — the square of the largest number is equal to the sum of the squares of the two smaller numbers. The multiple of any Pythagorean [more…]

### The Sine Function: Opposite over Hypotenuse

When you’re using right triangles (triangles with right angles in them) to define trig functions, the trig function *sine*, abbreviated *sin*, has input values that are angle measures and output values that [more…]