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Trigonometry functions - introduction

There are six functions that are the core of trigonometry. There are three primary ones that you need to understand completely:

  • Sine  (sin)
  • Cosine  (cos)
  • Tangent  (tan)

The other three are not used as often and can be derived from the three primary functions. Because they can easily be derived, calculators and spreadsheets do not usually have them.

  • Secant  (sec)
  • Cosecant  (csc)
  • Cotangent  (cot)

All six functions have three-letter abbreviations (shown in parentheses above).

Definitions of the six functions

Consider the right triangle on the left. For each angle P or Q, there are six functions, each function is the ratio of two sides of the triangle. The only difference between the six functions is which pair of sides we use.

In the following table

  • a is the length of the side adjacent to the angle (x) in question.
  • o is the length of the side opposite the angle.
  • h is the length of the hypotenuse.

"x" represents the measure of ther angle in either degrees or radians.

Sine The three primary functions
Cosine
Tangent
Cosecant Notice how each is the reciprocal of sin, cos or tan.
Secant
Cotangent

For example, in the figure above, the cosine of x is the side adjacent to x (labeled a), over the hypotenuse (labeled h): If a=12cm, and h=24cm, then cos x = 0.5  (12 over 24).

Soh Cah Toa

These 9 letters are a memory aid to remember the ratios for the three primary functions - sin, cos and tan. Pronounced a bit like "soaka towa".  See Sohcahtoa.

The ratios are constant

Because the functions are a ratio of two side lengths, they always produce the same result for a given angle, regardless of the size of the triangle.

In the figure above, drag the point C. The triangle will adjust to keep the angle C at 30°. Note how the ratio of the opposite side to the hypotenuse does not change, even though their lengths do. Because of that, the sine of 30° does not vary either. It is always 0.5.

Remember: When you apply a trig function to a given angle, it always produces the same result. For example tan 60° is always 1.732.

Using a calculator

Most calculators have buttons to find the sin, cos and tan of an angle. Be sure to set the calculator to degrees or radians mode depending on what units you are using.

Inverse functions

For each of the six functions there is an inverse function that works in reverse. The inverse function has the letters 'ARC' in front of it.

For example the inverse function of COS is ARCCOS. While COS tells you the cosine of an angle, ARCCOS tells you what angle has a given cosine. See Inverse trigonometric functions.

On calculators and spreadsheets, the inverse functions are sometimes written acos(x) or cos-1(x).

Trigonometry functions of large and/or negative angles

The six functions can also be defined in a rectangular coordinate system. This allows them to go beyond right triangles, to where the angles can have any measure, even beyond 360°, and can be both positive and negative. For more on this see Trigonometry functions of large and negative angles.

Identities - replacing a function with others

Trigonometric identities are simply ways of writing one function using others. For example, from the table above we see that This equivalence is called an identity. If we had an equation with sec x in it, we could replace sec x with
one over cos x if that helps us reach our goals. There are many such identities. For more see Trigonometric identities.

Not just right triangles

These functions are defined using a right triangle, but they have uses in other triangles too. For example the Law of Sines and the Law of Cosines can be used to solve any triangle - not just right triangles.

Graphing the functions

The functions can be graphed, and some, notably the SIN function, produce shapes that frequently occur in nature. For example see the graph of the SIN function, often called a sine wave, on the right. For more see

Pure audio tones and radio waves are sine waves in their respective medium.

Derivatives of the trig functions

Each of the functions can be differentiated in calculus. The result is another function that indicates its rate of change (slope) at a particular values of x. These derivative functions are stated in terms of other trig functions. For more on this see Derivatives of trigonometric functions. See also the Calculus Table of Contents.

Other trigonometry topics

Angles

Trigonometric functions

Solving trigonometry problems

Calculus