4.5 graphs of tangent, cotangent, secant, and cosecant
TRANSCRIPT
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4.5
Graphs of Tangent, Cotangent, Secant, and Cosecant
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Quiz: 4-4 2433sin5.0)( xxf
1. Vertical stretch/shrink1. Vertical stretch/shrink2. Horizontal stretch/shrink2. Horizontal stretch/shrink3. Horizontal translation (phase shift)3. Horizontal translation (phase shift)4. Vertical translation 4. Vertical translation
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What you’ll learn about
• The Tangent Function• The Cotangent Function• The Secant Function• The Cosecant Function
… and whyThis will give us functions for the remaining trigonometric ratios.
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UNIT CIRCLE
r = 1
3030ºº
AA
CC
6060ºº
x
y
BB
3030ººAA
CC
BB
16060ºº
1
3
01
00tan2tan
x
ytan
2
2
33
232
130tan
6tan
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UNIT CIRCLE
-60-60ºº
undefined 0/1)90tan( x
ytan
3)60tan(
-30-30ºº
6060ºº3030ºº00ºº 9090ºº
11
-90-90ºº
Vertical asymptoteVertical asymptote
-1-1
33)30tan(
00tan
3330tan
360tan
0190tan
Vertical asymptoteVertical asymptote
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UNIT CIRCLE
x
ytan
00ºº 9090ºº
11
-90-90ºº
-1-1
Pattern repeatsPattern repeats
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Your turn:Your turn:1.1. Domain = ?Domain = ?2.2. Range = ?Range = ?3.3. Continuous?Continuous?4.4. Symmetry?Symmetry?5.5. bounded?bounded?6.6. Vertical/horizontal asymptotes?Vertical/horizontal asymptotes?7.7. End behavior?End behavior?
Your turn:
0)(lim xfx
0)(lim xfx
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f(x) = sin x
-1.5
-1
-0.5
0
0.5
1
1.5
0 1 2 3 4 5 6 7
radians
radians
Remember this?Remember this?
The The y-coordinatey-coordinate of the graph is the of the graph is the distance above/belowdistance above/below the x-axis on the unit circle.the x-axis on the unit circle.
The The x-coordinatex-coordinate of the graph is the of the graph is the distance arounddistance around the unit circle. the unit circle.
Same thingSame thing for for cos x.cos x.
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How do we construct the Tangent Function’s graph?
Sin xSin x
Cos xCos x
y = Sin xy = Sin x: : the verticalthe vertical distance above/belowdistance above/below the x-axis (on the unit circle).the x-axis (on the unit circle).
x = cos xx = cos x: : the horizontalthe horizontal distance to the right/left ofdistance to the right/left of the y-axis (on the unit circle).the y-axis (on the unit circle).
This distance is graphed asThis distance is graphed as the y-coordinate on this graph.the y-coordinate on this graph.
This distance is also graphed asThis distance is also graphed as the y-coordinate on this graph.the y-coordinate on this graph.
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Construct the Tangent Function using the graph of ‘y’ and ‘x’
)(
)(tan
lengthadjacent
lengthoppositeA
x
yA tan
y = Sin xy = Sin x
x = Cos xx = Cos x
y = Sin xy = Sin x: (on unit circle): (on unit circle)
x = cos xx = cos x: (on unit circle): (on unit circle)
Using the graphs of the left:Using the graphs of the left:
1
00tan
0
190tan
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The Tangent and Cotangent Functions
Tan xTan x
Cot xCot x
tan
1cot
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Asymptotes and Zeros of the Tangent Function
x
xx
cos
sintan
b
Rules for Tangent: y = tan (bx – c) + d
a = none
period =
A full period of tan occurs between 2 consecutive asymptotes.
To find 2 consecutive asymptotes: bx – c = and bx – c =
The graph starts low, ends high, ½ way thru period is where it crosses the x-axis.
X-intercepts: sin X-intercepts: sin x x = 0= 0
Asymptotes: cos Asymptotes: cos xx = 0 = 0
2
b
2
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The Tangent Function
x
xx
cos
sintan
2
X-intercepts: sin X-intercepts: sin x x = 0= 0
Asymptotes: cos Asymptotes: cos xx = 0 = 0
2
b
2
AmplitudeAmplitude: doesn’t have one (look at the graph): doesn’t have one (look at the graph)
dcbxxf )tan()(
PeriodPeriod: : A full period of tan(x) occurs between 2 A full period of tan(x) occurs between 2 asymptotes which occur at asymptotes which occur at oddodd multiples of multiples of
Shrink factor causes asymptotes to be odd multiplies of: Shrink factor causes asymptotes to be odd multiplies of: b2
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Describe how tan(x) is transformed to graph: -tan(2x)
dcbxxf )tan()(
b
2
1. 1. AmplitudeAmplitude: n/a: n/a
2. 2. PeriodPeriod: :
3. 3. Vertical AsymptotesVertical Asymptotes: odd multiples of: odd multiples of b2
4
4. -1 coefficient: causes reflection across x-axis.4. -1 coefficient: causes reflection across x-axis.
Tan(x)Tan(x) -tan(2x)-tan(2x)
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The Sine and Cosecant Functions
sin
1csc
Sine (x)Sine (x)
Cosecant (x)Cosecant (x)
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The Cosecant Function: y = 2 csc 2x
To graph graphing shifts, rewrite the equation as it’s reciprocal, then find zero’s, max/mins (amp), and period.
dcbxxf
)sin(
1)(
1. Vertical stretch: factor of 21. Vertical stretch: factor of 2
)2sin(
2)(
xxf
sin(x) csc(x) sin(x) csc(x)
2. Vertical asymptotes: even multiples of2. Vertical asymptotes: even multiples of .,2*2,2*02
etc
3. Horizontal shrink factor of 3. Horizontal shrink factor of ½, changes period to ½, changes period to 2csc(2x)2csc(2x)
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The Cosine and Secant Functions
Cos xCos x
cos
1sec
Sec xSec x
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Solve Trig function AlgebraicallyFind: ‘x’ such that and sec(x) = -2 Find: ‘x’ such that and sec(x) = -2 2
3 x
2)sec( x
rx
1
22
x
r
--½½
11
23
Converting to unit circle:Converting to unit circle:
r = 1 x = -r = 1 x = -½½
3
3030ºº
1 6060ºº½
2
That looks familiar!That looks familiar!
6060ºº
240240ºº x = 240x = 240ºº 34
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Solving a Trig equation graphically
Find the smallest number ‘x’ such that:Find the smallest number ‘x’ such that: xx csc2 1. Graph and 1. Graph and
2xy )csc(xy
2. Find the point of intersection2. Find the point of intersection
068.1x
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Basic Trigonometry Functions
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HOMEWORK
Section 4-5