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Tangent and Secant Identities on a Unit Circle

Starting with the Pythagorean identity, sin2θ + cos2θ = 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…]

Dealing with Half-Angle Identities Involving Radicals

By adding, subtracting, or doubling angle measures, you can find lots of exact values of trigonometry functions. For example, you can use the half-angle identity when the exact value of the trig function [more…]

How to Find a Common Denominator of a Fraction to Solve a Trig Identity

Fractions are your friends. You may not believe this now, but the more you work with trigonometry functions, the more you’ll like fractions. Finding a common denominator to combine fractions often paves [more…]

How to Multiply by a Conjugate to Find a Trigonometry Identity

Conjugates offer a great way to find trigonometry identities. In mathematics, a conjugate consists of the same two terms as the first expression, separated by the opposite sign. For instance, the conjugate [more…]

How to Square Both Sides to Solve a Trigonometry Identity Problem

When you work on both sides of a trig identity at the same time, you may sometimes need to square both sides. You generally use this technique when one side or the other [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…]


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