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Precalculus: Trigonometric Functions

Trigonometric Functions

Problems

Trigonometric Functions, page 2

page 1 of 2

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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