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Trigonometry Basics

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How to Use SohCahToa to Find the Trig Functions of a Right Triangle

The study of trigonometry begins with the right triangle. The three main trig functions (sine, cosine, and tangent) and their reciprocals (cosecant, secant, and cotangent) all tell you something about [more…]

How to Graph Sine, Cosine, and Tangent

So, you need to graph a sine, cosine, or tangent function. Sine, cosine, and tangent — and their reciprocals, cosecant, secant, and cotangent — are periodic [more…]

How to Measure Angles with Radians

Degrees aren’t the only way to measure angles. You can also use radians. Degrees and radians are just two different ways to measure angles, like inches and centimeters are two ways of measuring length. [more…]

Even-Odd Identities in Trigonometric Functions

All functions, including trig functions, can be described as being even, odd, or neither. Knowing whether a trig function is even or odd can help you simplify an expression. These even-odd identities are [more…]

How to Determine the Length of an Arc

Knowing how to calculate the circumference of a circle and, in turn, the length of an arc — a portion of the circumference — is important in pre-calculus because you can use that information to analyze [more…]

How to Graph a Cosine Function

The parent graph of cosine looks very similar to the sine function parent graph, but it has its own sparkling personality (like fraternal twins). Cosine graphs follow the same basic pattern and have the [more…]

How to Calculate an Angle Using Reciprocal Trigonometric Functions

Three trigonometric ratios — secant, cosecant, and cotangent — are called reciprocal functions because they're the reciprocals of sine, cosine, and tangent. These three functions open up three more ways [more…]

How to Calculate an Angle Using Inverse Trigonometric Functions

Almost every function has an inverse. An inverse functionbasically undoes a function. The trigonometric functions sine, cosine, and tangent all have inverses, and they're often called [more…]

How to Draw Uncommon Angles

Many times in your journey through trigonometry — actually, all the time — drawing a figure will help you solve a given problem. So what do you do if you're asked to draw an angle that has a measure greater [more…]

How to Work with 30-60-90-Degree Triangles

All 30-60-90-degree triangles have sides with the same basic ratio. If you look at the 30–60–90-degree triangle in radians, it translates to the following: [more…]

How to Calculate the Sine of an Angle

Because you spend a ton of time in pre-calculus working with trigonometric functions, you need to understand ratios. One important ratio in right triangles is the sine. The [more…]

How to Calculate the Cosine of an Angle

Because you spend a ton of time in pre-calculus working with trigonometric functions, you need to understand ratios. One important ratio in right triangles is the cosine. The [more…]

How to Find the Sine of a Doubled Angle

You use a double-angle formula to find the trig value of twice an angle. Sometimes you know the original angle; sometimes you don't. Working with double-angle formulas comes in handy when you're given [more…]

How to Find the Tangent of a Doubled Angle

The double-angle formula for tangent is used less often than the double-angle formulas for sine or cosine; however, you shouldn't overlook it just because it isn't as popular as its cooler counterparts [more…]

How to Use Half-Angle Identities to Evaluate a Trig Function

You can use half-angle identities to evaluate a trig function of an angle that isn't on the unit circle by using one that is. For example, 15 degrees, which isn't on the unit circle, is half of 30 degrees [more…]

How to Express Products of Trigonometric Functions as Sums or Differences

If you can break up a product of trig functions into the sum of two different terms, each with its own trig function, doing the math becomes much easier. In pre-calculus, problems of this type usually [more…]

How to Express Sums or Differences of Trigonometric Functions as Products

It's a good idea to familiarize yourself with a set of formulas that change sums to products. Sum-to-product formulas are useful to help you find the sum of two trig values that aren't on the unit circle [more…]

How to Eliminate Exponents from Trigonometric Functions Using Power-Reducing Formulas

Power-reducing formulas allow you to get rid of exponents on trig functions so you can solve for an angle's measure. This ability comes in very handy in calculus. [more…]

How to Prove an Equality Using Reciprocal Identities

Oftentimes, your math teachers will ask you to prove equalities that involve the secant, cosecant, or cotangent functions. Whenever you see these functions in a proof, the reciprocal identities usually [more…]

How to Simplify an Expression Using Even/Odd Identities

Because sine, cosine, and tangent are functions (trig functions), they can be defined as even or odd functions as well. Sine and tangent are both odd functions, and cosine is an even function. In other [more…]

How to Simplify an Expression Using Co-function Identities

If you take the graph of y = sin x and shift it to the left by pi/2 units, it looks exactly like the graph of y= cos x. The same is true for tangent and cotangent, as well as secant and cosecant. That's [more…]

How to Prove an Equality by Using Periodicity Identities

Using the periodicity identities comes in handy when you need to prove an equality that includes the expression (x + 2pi) or the addition (or subtraction) of the period. For example, to prove [more…]

How to Prove Trigonometric Identities When You Start Off with Fractions

When the trig expression you're given begins with fractions, most of the time you have to add (or subtract) them to get things to simplify. Here's one example of a proof where doing just that gets the [more…]

How to Simplify Trigonometric Expressions with a Binomial in a Fraction's Denominator

When a trigonometric expression is a fraction with a binomial in its denominator, always consider multiplying by the conjugate before you do anything else. Most of the time, this technique allows you to [more…]

How to Prove Complex Identities by Working Individual Sides of a Trig Proof

Sometimes doing work on both sides of a trig proof, one side at a time, leads to a quicker solution. This is because in order to prove a very complicated identity, you may need to complicate the expression [more…]

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