Analytic Trigonometry

Barnett Ziegler Bylean

CHAPTER 3Graphs of trig functions

CH 1 - SECTION 1Basic graphs

Why study graphs?

Assignment

• Be able to sketch the 6 basic trig functions WITHOUT referencing notes or using a graphing calculator.

• Be able to answer questions concerning: domain/range x-int/y=int increasing/decreasing symmetry asmptote without notes or calculator.

Hints for hand graphs

• X-axis - count by π/2 with domain [-2π, 3π]• Y-axis – count by 1’s with a range of [-5,5]

Defining trig functions in terms of (x,y)Input x

Output cos()=, sin(x)=, tan(x) ,sec(x), csc(x), cot(x)

ө(cos(ө),sin(ө) )

y=sin(x)

• Domain/range• X-intercept• Y-intercept• Other points• Periodic/period• Increase• Decrease• Symmetry (odd)

x

y

Input x

Output cos()=,sin(x)= , tan(x) ,sec(x), csc(x), cot(x)

ө(cos(ө),sin(ө) )

x

y

Output y = sin(x) – using π/2 for the x-scale

x

y

x

y

y = cos(x)Input x

Output cos()=ө

(cos(ө),sin(ө) )

Domain/rangeX-interceptY-interceptOther pointsPeriodic/periodIncreaseDecreaseSymmetry (odd)

x

y

x

y

x

y

x

y

y = tan(x) and y = cot(x)Input x

ө(cos(ө),sin(ө) ) y = tan(x) y = cot(x)

restricted/asymptotes: Range ? y-intercept x-intercept

x

y

x

y

y = sec(x) and y = csc(x)Input x

sec(x) csc(x) restricted/asymptotes? range?ө

(cos(ө),sin(ө) )

x

y

x

y

CHAPTER 3 – SECTION 2Transformations of sin and cos

Review transformations

• Given f(x)• What do you know about the following• f(x-3) f(x + 5)• f(3x) f(x/7)

• f(x) + 6 f(x) – 4• 3f(x) f(x)/3

Trigonometric Transformations - dilations

• Y = Acos(Bx) y = Asin(Bx)• Multiplication causes a scale change in the

graph• The graph appears to stretch or compress

Vertical dilation : y = Af(x)

• If the multiplication is external (A) it multiplies the y-co-ordinate (stretches vertically) – the x intercepts are stable (y=0), the y intercept is not stable for cosine

• The height of a wave graph is referred to as the amplitude (direct correlation to physics wave theory) - It is how much impact the x has on the y value - louder sound, harder heartbeat etc.

• Amplitude is measured from axis to max. and from axis to min.

Examples of some graphs

•

x

y

x

y

x

y

y=3(sin(x)

y=-2sin(x)

y=sin(x)𝜋2

y = 3cos(x)

x

y

Scale π/21

Horizontal Dilations

• If the multiplication is inside the function it compresses horizontally against the y-axis – the x-intercepts are compressed – the y- intercept is stable – this affects the period of the function

• Period – the length of the domain interval that covers a full rotation – The period for sine and cosine is 2π – multiplying the x – coordinates speeds up the rotation thereby compressing the period -

• New period is 2π /multiplier• Frequency – the reciprocal of the period-

Examples of some graphs

y=cos(x)

x

y

y=cos(2x)x

y

x

y

y= cos(x/2)

Sketch a graph (without a calculator)

• y = 3cos(2x)

• y = - sin(πx)

Transformations - Vertical shifts

• Adding “outside” the function shifts the graph up or down – think of it like moving the x-axis

• f(x) = sin(x) + 2 g(x) = cos(x) - 4

Pertinent information affected by shift

• the amplitude and period are not affected by a vertical shift

• The x and y intercepts are affected by shift – • The maximum and minimum values are

affected by vertical shift

Finding max/min values

• Max/min value for both sin(x) and cos(x) are 1 and -1 respectively

• Amplitude changes these by multiplying• Shift change changes them by adding• Ex: k(x)= 4cos(3x -5) – 2• the max value is now 4(1)- 2= 2• the min value is now 4(-1) – 2 =-6

Example

• Graph k(x) = 4 + 2cos(𝛑x)

Summary

•

Writing equations

• Identify amplitude• Identify period• Identify axis shift

CHAPTER 3 – SECTION 3Horizontal shifts

Simple Harmonics

• f(x) = Asin(Bx + C) or g(x) = Acos(Bx + C) are referred to as Simple Harmonics.

• These include horizontal shifts referred to as phase shifts

• The shift is -C units horizontally followed by a compression of 1/B - thus the phase shift is

-C/B units• The amplitude and period are not affected by

the phase shift

Horizontal shift

• f(x) = cos(x + )

• g(x) = cos(2x – )

find amplitude, max, min, period and phase shift• f(x) = 3cos(2x – π/3) • y = 2 – 4sin(πx + π/5)

CHAPTER 3 – SECTION 6Tangent/cotangent/secant/cosecant revisited

Basic graphs

• asymptotes• Period• Increasing/decreasing

• tan(x) cot(x)

• sec(x) csc(x)

k + A tan(Bx+C) or k + A cot(Bx+C)

• No max or min - effect of A is minimal• Period is π/B instead of 2π/B• Phase shift is still -C/B and affects the x intercepts and asymptotes• k moves the x and y intercepts

Examples

• y = 3 + 2tan(3x)

• y = cot()

k+ Asec(Bx+ C) or k + Acsc(Bx + C)

• local maxima and minima affected by k and A

• Directly based on sin and cos so Period is 2π/B

• Shift is still -C/B

Examples

• y = 3 + 2sec(3πx)

• y = 1 – csc (2x + π/3)