Analytic Trigonometry

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# Analytic Trigonometry - PowerPoint PPT Presentation

Analytic Trigonometry . Barnett Ziegler Bylean. Graphs of trig functions. Chapter 3. Basic graphs. Ch 1 - section 1. Why study graphs?. Assignment. Be able to sketch the 6 basic trig functions WITHOUT referencing notes or using a graphing calculator.

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### Analytic Trigonometry

Barnett Ziegler Bylean

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

(cos(ө),sin(ө) )

ө

Output cos()=,

sin(x)=,

tan(x) ,sec(x), csc(x), cot(x)

y=sin(x)

Input x

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

(cos(ө),sin(ө) )

ө

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

Output

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

y = cos(x)

Input x

(cos(ө),sin(ө) )

Output cos()=

ө

Domain/range

X-intercept

Y-intercept

Other points

Periodic/period

Increase

Decrease

Symmetry (odd)

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

Input x

y = tan(x) y = cot(x)

restricted/asymptotes:

Range ?

y-intercept

x-intercept

(cos(ө),sin(ө) )

ө

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

Input x

sec(x) csc(x)

restricted/asymptotes?

range?

(cos(ө),sin(ө) )

ө

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

y=3(sin(x)

y=sin(x)

y=-2sin(x)

y = 3cos(x)

1

Scale π/2

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(2x)

y= cos(x/2)

y=cos(x)

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)
Writing equations
• Identify amplitude
• Identify period
• Identify axis shift
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)
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)