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Trig Equations and Applications

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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 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…]

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…]

How to Work with Inverse Trigonometry Functions

The easiest way to work with inverse trig functions is to have a chart handy with the exact values of the functions. When angles other than the most common or popular are involved, you can either use a [more…]

Toggle between Radians and Degrees on Your Scientific Calculator

Scientific calculators are very accommodating — they give you results in either degrees or radians, depending on which mode you set them in. This feature is great, but it trips up even the best mathematicians [more…]

Use the Inverse Trigonometry Function or Inverse of a Reciprocal Function on Your Calculator

On scientific calculators, the –1 or x1 button means to find the reciprocal of a number. This reciprocal button allows you to find the value of a reciprocal function when you’re working with a number. [more…]

How to Solve Inverse Trigonometry Functions with Uncommon Angles

When working with inverse trig functions, it’s always more convenient when the numbers you’re working with are the results of applying one of the trig functions to a common angle measure. When the angle [more…]

Solve Trigonometry Equations by Factoring

The same type of factoring that algebra uses to solve equations is a great help in solving trigonometry equations. The only trick with the trig equations is to recognize that instead of just [more…]

Solve a Trig Equation by Finding a Greatest Common Factor

The trigonometry equations that require finding a greatest common factor, factoring it out, and then solving the equation could look like one of the following two equations: [more…]

How to Factor Trigonometry Expressions by Grouping

The process of factoring by grouping works in very special cases, when the original trigonometry expression is the result of multiplying two binomials together that have some unrelated terms in them. You [more…]

How to Solve a Trigonometry Equation Using the Quadratic Formula

When trigonometry quadratic equations factor, life is good. When they don’t, you can still survive, thanks to that wonderful quadratic formula. In case you’ve forgotten the exact formula, here it is. [more…]

How to Solve a Trig Equation That Has Multiple Trigonometry Functions

Some trigonometry equations contain more than one trig function. Others have mixtures of multiple angles and single angles with the same variable. Some examples of such equations include 3cos [more…]

How to Find a Solution to a Multiple-Angle Trig Equation

Multiple-angle trigonometry expressions are those where the angle measure is some multiple of a variable — for example, 2x or 3y. You can take these expressions apart and solve for all the additional solutions [more…]

How to Square Both Sides of a Trig Equation

When solving trigonometry equations, more than one method will usually work — although one method is often quicker or easier than another. With practice, you’ll get good at choosing the better of the ways [more…]

How to Multiply through a Trig Equation with Another Function

The technique of multiplying through a trigonometry equation by a carefully selected function shouldn’t be your first choice — or your second, third, or fourth choice. This method is usually a last resort [more…]

How to Solve a Trig Equation Using a Graphing Calculator

Some of the more-advanced graphing calculators make short work of solving trigonometry equations. A graphing calculator comes in very handy when the equation is complicated, has several different functions [more…]

How to Use the Law of Sines with a Triangle

When you already know two angles in a triangle as well as one of the sides, such as in the case of ASA or AAS, you can use the law of sines to find the measures of the other two sides. This law uses the [more…]

Use the Law of Cosines with a Triangle

The law of cosines comes in handy when you have two or more sides of a triangle — as in situations involving SSS and SAS — and need to use trigonometry to find the measures of the other three parts. When [more…]

Use the Law of Cosines for SAS

When you have two sides of a triangle and the angle between them, otherwise known as SAS (side-angle-side), you can use the law of cosines to solve for the other three parts. Consider the triangle [more…]

Use the Law of Cosines with SSS

When you know the values for two or more sides of a triangle, you can use the law of cosines. In the following case, you know all three sides (which is called SSS, or side-side-side, in trigonometry) but [more…]

Use the Law of Cosines for SSA

The Law of Cosines works well for solving triangles when you have two sides and an angle, but the angle isn’t between the two sides. In this case, the Law of Sines isn’t an option. Also, to solve a triangle [more…]

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