The main three trigonometric functions are the sine function, cosine function, and tangent function, sometimes represented as sin x, cos x, and tan x. These functions measure the ratio between different sides of a triangle. The easiest way to memorize these trigonometric functions is the mnemonic device SOHCAHTOA:

**S**ine = **O**pposite over **H**ypotenuse

**C**osine = **A**djacent over **H**ypotenuse

**T**an = **O**pposite over **A**djacent

Understanding tangent, sine, and cosine as functions enables us to figure out the range and domain of tangent, sine, and cosine.

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## Reciprocal vs. Inverse Trigonometric Functions

There are also three reciprocal functions. The reciprocal of a function is when the numerator and denominator of a fraction or ratio are flipped. The inverse is the “opposite” or “reversal” of the function.

The cosecant function, or csc x, is the reciprocal of sin x (not to be confused with arcsin, the inverse of the inverse sine function).

The secant function, or sec x, is the reciprocal of cos x (not to be confused with arccos or arccosine, the inverse cosine function).

The cotangent function, or cot x, is the reciprocal of tan x (not to be confused with arctan or arctangent, the inverse tangent function).

Cosecant = Hypotenuse over opposite

Secant = Hypotenuse over adjacent

Cotan = Adjacent over opposite

## Finding the Range and Domain of Tangent, Sine, and Cosine

You can graphically represent all of the trigonometric functions. If you give each function an angle as input (the domain is the possible range of values for the input), you will get an output value (the range).

Here is the graph of the sine function:

In the sine function, the domain is all real numbers and the range is -1 to 1.

Here is the graph of the cosine function:

This has the same domain and range as the last graph. Again, the domain is all real numbers, and the range is -1 to 1.

Here is the graph of the tangent function:

The domain is all numbers **except for**

*x*. The range is all real numbers.

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## Finding the Sine, Cosine, and Tangent Domains

The sine, cosine, and tangent functions are all functions that can be graphed. Therefore, they all have bounds to the possible range of values for their x-value (domain) and y-value (range). Sine and cosine both have domains of all real numbers. Tangent is the one whose domain is limited to all values except for

plus any repeating value of .