Function and Relation Library

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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( ).

Open this page & move point A
with the mouse to see all 6 functions move.

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 , 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 = - csc²(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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