Ch. 5 Ch. 5 Trigonometric Trigonometric Functions of Functions of

Real NumbersReal NumbersMelanie KuleszMelanie KuleszKatie LariviereKatie Lariviere

Austin WittAustin Witt

The circle of radius 1 centered at the origin in the xy-plane.

x2 + y2 = 1

Proving points on the unit Proving points on the unit circlecircle

Use equation: x2 + y2 = 1Use equation: x2 + y2 = 1 See ExampleSee Example

See example Use equation: x2 + y2

= 1

Terminal PointsTerminal PointsTerminal Point – the point Terminal Point – the point PP((xx,,yy) obtained ) obtained

and determined by the real number and determined by the real number ttSuppose Suppose tt is a real number. Mark off a is a real number. Mark off a

distance distance tt along the unit circle, starting at along the unit circle, starting at the point (1,0) and moving in a the point (1,0) and moving in a counterclockwise direction if counterclockwise direction if t t is positive or is positive or in a clockwise direction if in a clockwise direction if tt is negative is negative

See example See example tt = = --ππ

Reference NumbersReference Numbers Reference Number - the shortest distance Reference Number - the shortest distance

along the unit circle between the terminal point along the unit circle between the terminal point determined by determined by tt and the x-axis and the x-axis

Sine CurveSine Curve Cosine CurveCosine Curve Tangent CurveTangent Curve StretchStretch ShiftShift AmplitudeAmplitude PeriodPeriod

To find the terminal point P determined by any value of t, use the following steps…

1.Find the reference number t2.Find the terminal point Q(a, b)

determined by t3.The terminal point determined by t is

P(±a, ±b ), where the signs are chosen according to the quadrant in which this terminal point lies

See Example

Trigonometric Functions

sin t = ycos t = xtan t = y/x (x≠0)csc t = 1/y (y≠0)sec t = 1/x (x≠0)cot t = x/y (y≠0)See example

Even-Odd Properties Sin(-t) = -sin t Cos(-t) = cos t Tan(-t) = -tan t

Csc(-t) = -csc t Sec(-t) = sec t Cot(-t) = -cot t

Even

Odd

SIGNS OF THE TRIGONOMETRIC SIGNS OF THE TRIGONOMETRIC FUNCTIONSFUNCTIONS

Quadrant Positive Functions Negative functions

I all noneII sin, csc cos, sec, tan, cotIII tan, cotsin, csc, cos, secIV cos, sec sin, csc, tan, cot

Fundamental IdentitiesFundamental Identities● ● Reciprocal Identities:Reciprocal Identities:

csc csc tt = 1/sin = 1/sin tt sec sec tt = 1/cos = 1/cos ttcot cot tt = 1/tan = 1/tan tt tan tan tt = sin = sin tt/cos /cos ttcot cot tt = cos = cos tt/sin /sin tt

● ● Pythagorean Identities:Pythagorean Identities:sin^2sin^2tt + cos^ + cos^tt = 1 = 1tan^2tan^2tt + 1 = sec^2 + 1 = sec^2tt1 + cot^21 + cot^2tt = csc^2 = csc^2tt

Trigonometric Graphs• Periodic Properties:

The functions tan and cot have period πtan(x + π) = tan xcot(x + π) = cot xThe functions csc and sec have period 2 πcsc(x + 2π) = csc xsec(x + 2π)= sec x

• The functions of Sine and Cosine both have a period of 2π

• This means they repeat themselves after one full rotation around the unit circle

• The sine function starts from the origin• It then follows the pattern of Peak, Root, Valley• The roots are at every 1 Pi when the period is 2 Pi• The peaks are equal to the amplitude which is equal to the

coefficient of the function.• Valleys are also derived from the amplitude

F(x)=sinx

The Function of CosineThe Function of Cosine• The Cosine Function starts at a peak which is equal to The Cosine Function starts at a peak which is equal to • amplitude or coefficient of the function.amplitude or coefficient of the function.• It then follows the pattern root, valley, peak. The rootsIt then follows the pattern root, valley, peak. The roots• occurring at every 1/2Pi. occurring at every 1/2Pi. • The valleys and peaks equal to the amplitude.The valleys and peaks equal to the amplitude.

Horizontal stretching occurs when you a have a change of

the period of the function. Ex 1. sin2x would repeat itself twice in the one

rotation of the unit circle. Ex2. sin1/2x would repeat itself once in 2 rotations of

the unit circle. Vertical stretching occurs from a change in amplitude

or the coefficient of function. Ex 1. 2sinx would have a peak and valley at 2 and -2

respectively.

Horizontal shifts of the sine and cosine functions are shown as sin(x+a)

where is some value in radians.

Vertical shifts look like sinx+a which would move it up or down depending on (a).

The Tangent FunctionThe Tangent Function• The tangent function has a period of Pi but

starts out at negative ½ Piand goes to positive ½ Pi.• Its shape liked an “s” and intersects the

origin in the middle• It also has asymptotes' at the beginning and

end of each period

Cotangent = 1/tan : the reciprocal of tangent starts at the origin with an asymptotes at the origin and has a period of 1 Pi where it ends with

another asymptote. It too looks like an “s” but it has a negative slope as it moves

from Positive infinity to negative infinity in its “Y” values.

Cosecant =1/sin : the reciprocal of sine has asymptotes at every ½ Pi . If you take the peaks of the cosine function that is the vertex of the Parabola formed by the reciprocal

Secant= 1/cos: the reciprocal of the cosine function is related to the Cosecant function in that its parent function’s peaks are the vertices of the Parabolas formed. However secant has asymptotes at 0 and 1 Pi instead Of every ½ pi.