# Precalculus: Trigonometric Functions

## Contents

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#### Trigonometric Functions

There are three basic ways to define the trigonometric functions.

Consider a point (x, y) on the terminal side of an angle θ in standard position. It lies a distance d away from the origin.

trigfunctions

 sine(θ) = sin(θ) = .

 cosine(θ) = cos(θ) = .

 tangent(θ) = tan(θ) = .

 cosecant(θ) = csc(θ) = .

 secant(θ) = sec(θ) = .

 cotangent(θ) = cot(θ) = .

The unit circle is the circle with equation x 2 + y 2 = 1 . Let s be the length of the arc with one endpoint at (1, 0) extending around the circle counterclockwise with its other endpoint at (x, y) . Note that s is both the length of an arc as well as the measure in radians of the central angle that intercepts that arc.

trigfunctions2

 sine(s) = sin(s) = y.

 cosine(s) = cos(s) = x.

 tangent(s) = tan(s) = ;;x≠0.

 cosecant(s) = csc(s) = ;;y≠0.

 secant(s) = sec(s) = ;;x≠0.

 cotangent(s) = cot(s) = ;;y≠0.

Consider a right triangle with one acute angle θ in standard position. Let the side opposite that angle be called the opposite side. Let the other leg be called the adjacent side.

trigfunctions3

 sine(θ) = sin(θ) =

 cosine(θ) = cos(θ) =

 tangent(θ) = tan(θ) =

 cosecant(θ) = csc(θ) =

 secant(θ) = sec(θ) =

 cotangent(θ) = cot(θ) =

The trigonometric functions have different signs according to the quadrant in which the angle's terminal side lies. Here is a chart to show these signs.

Figure %: The signs of the trigonometric functions in the quadrants

The quadrantal angles have the values as shown in the chart below.

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