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Precalculus: Trigonometric Functions

Trigonometric Functions

Problems

Problems

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.

Figure %: The values of the trigonometric functions of quadrantal angles

A reference angle is the positive acute angle created by the terminal side of an angle in standard position and the x -axis. A reference angle, therefore, is a first quadrant angle. Let β be the reference angle of θ . The value of a trigonometric function at θ is equal to the value of that function at β -- unless there is a variation in sign. The sign difference depends on which quadrant θ is in. An understanding of reference angles simplifies the evaluation of trigonometric functions of large angles.

Figure %: In each drawing, β is the reference angle for θ .
Figure %: How to calculate the reference angle β for any angle θ between 0 and 2Π radians.

It will be useful to memorize the values of the trigonometric functions at a few common angles besides the quadrantal angles: θ = 30 o , 45 o , 60c irc . Use the values below to find the values of cosecant, secant, and cotangent at these angles.

values

sin(30 o ) = .    

cos(30 o ) = frac 2.    

tan(30 o ) = .    

sin(45 o ) = .    

cos(45 o ) = frac1.    

tan(45 o ) = 1.    

sin(60 o ) =    

cos(60 o ) = frac12    

tan(60 o ) =    

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