 ## Function and Relation Library

Trigonometric Functions:     Angle Definitions     Legs of A Triangle Definitions
Sine     Cosine     Tangent     Secant     Cosecant     Cotangent

Trig functions are functions of a number, y = f(x) and very often that number is an angle, y = f(x) or y = f( ).   Visit the Interactive Sketch Pad MaterialOn Trig Functions sinplay   -- just right triangle sine cosplay   -- just right triangle cosine tanplay   -- just right triangle tangent TraceSin   -- drag point f(x) to state and show new ( x, f(x)) TraceCos   -- drag point f(x) to state and show new ( x, f(x)) TraceTan   -- just graph with a slideable point to display coordinates SinCosTan1stDay   -- Just sine, cosine, tangent of right triangle angles. SinCosTan6fx   -- All 6 functions in right triangle only. SinCosTanAllNumbers   -- All 6 functions in right triangle and forall numbers,     * controlled by figure for right triangle,     * parameter-defined angle measure in degrees for all measures, and     * parameter-defined angle measure in decimal number times pi for radian measures sinNcos.gsp   -- sine and cosine in right triangle only 2 Sine Functions -- Asin(Bx-C) + D on 2 functions for comparison also h(x) = sine + cosine SinTraceDerSums -- Sine from many views, right triangle, ABCD, derivative, antiderivative

As the point on the circle changes, the angle changes, since the point is on the terminal side of the angle.     As the angle changes, the length of each of the segments it determine changes.     As the length of the segment changes, the trig function changes.     Open the page in the box above to see this work!

• The sine, is the length of AE.
• The cosine is the length of BE.
• The tangent is the length of FC.
• The cosecant is the length of BG.
• The secant is the length of BF.
• The cotangent is the length of DG.

For a radius other than one, the ratios below must be used to scale up or down the size of the circle and yield the value of the function.

For more on this read The Unit Circle. Functions may also be defined in terms of x and y and r of right triangle ABE.     The horizontal segment BE is leg x.     The vertical segment AE is leg y.     The segment AB is side r and the hypotenuse of the right triangle and is still the radius of the circle on which point (x,y) lies, and still the distance from the point to the origin.     Defined in terms of x and y and r -- the lengths of BE, AE, and AB -- the sides of right triangle ABE, here are the trig functions. The sine is the ratio AE/AB. The cosine is the ratio BE/AB. The tangent is the ratio AE/BE. The cosecant is the ratio AB/AE. The secant is the ratio AB/BE. The cotangent is the ratio BE/AE.     For info on each function, read below.

THE SINE FUNCTION
• y = sin (x)
• opposite function: y = - sin(x)
• reciprocal function: y = csc (x), the cosecant
• inverse function: y = arcsin (x), the arcsine
• slope function: y = cos(x), the cosine
• period: 2 or 360°
• range: -1 < y < 1 The sine is the ratio of the y to r, the ratio of the vertical component to the radius.

The sine is useful for describing natural phenomena and for writing Fourier series to describe relations.

The sine of x may be computed to desired accuracy using sin x = x - x3/3! + x5/5! - x7/7! + ...

THE COSINE FUNCTION
• y = cos(x)
• opposite function:y = - cos(x)
• reciprocal function: y = sec(x), the secant
• inverse function: y = arccos(x), the arccosine
• slope function: y = - sin(x)
• period: 2 or 360°
• range: -1 < y < 1 The cosine, cos x, is the ratio of x to r, the ratio of the horizontal component to the radius.

It is useful in describing natural phenomena particularly in writing Fourier series for relations.

The cosine of a number, cos(x), may be evaluated to desired accuracy using the expression cos(x) = 1 - x2/2! + x4/4! - x6/6! + ...

THE COSECANT FUNCTION
• y = csc(x)
• infinite or undefined: when x is 0 ± k( /2) where k= 1, 2, 3, ..., whenever the sine is 0
• opposite function: y = - csc(x)
• reciprocal function: y = sin(x), the sine
• inverse function: y= arccsc(x), the arccosecant
• slope function:y = -csc(x) · cot(x), the opposite of the product of the cosecant and cotangent.
• period: 2 or 360°
• range: - to -1 and also + 1 to + , the range does not include numbers between -1 and +1 The cosecant, csc x, is the ratio of r to x. It is the reciprocal of the sine.

Because it is the reciprocal of the sine, when the sine increases the cosecant decreases. When the sine reaches a maximum, the cosecant reaches a minimum.

When the sine is 0, the cosecant is undefined or infinite in size.

THE SECANT FUNCTION
• y = sec(x)
• infinite or undefined: when x is ± k( /2) where k= 1, 2, 3, ..., whenever the cosine is 0
• opposite function: y = - sec(x)
• reciprocal function: y = cos(x), the cosine
• inverse function: y = arcsec(x), the arcsecant
• slope function: y = sec(x) · tan(x), the product of the secant and tangent.
• period: 2 or 360°
• range: - to -1 and also + 1 to , the range does not include numbers between -1 and +1 The secant, sec x, is the reciprocal of the cosine, the ratio of r to x.

When the cosine is 0, the secant is undefined. When the cosine reaches a relative maximum, the secant is at a relative minimum.

THE TANGENT FUNCTION
• y = tan(x)
• infinite or undefined: when x is 0 ± k( ) where k= 1, 2, 3, ...
• opposite function: y = - tan(x)
• reciprocal function: y = cot(x), the cotangent
• inverse function: y = arctan(x), the arctangent
• slope function: y = sec²(x), the square of the secant
• period: or 180°
• range: - to + , all numbers are in the range The tangent is the ratio of y to x, the ratio of the sine to the cosine: tan(x) = sin(x)/cos(x).

The tangent is very useful in trigonometry.

THE COTANGENT FUNCTION
• y = cot(x)
• undefined: when x is ± k( /2) where k= 1, 2, 3, ..., whenever the tangent is 0
• opposite function: y = - cot(x)
• reciprocal function: y = tan(x), the tangent
• inverse function: y= arccot(x), the arccotangent
• slope function: y = - csc˛(x)
• period: or 180°
• range: - to + , all numbers are in the range The cotangent is the reciprocal of the tangent. It is the ratio of x to y. It is also the ratio of the cosine to the sine: cot(x) = cos(x)/sin(x).

This page is from Exploring Functions Throught the Use of Manipulatives (ISBN: 0-9623593-3-5).

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