Chapter 7: Trigonometric Functions

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Chapter 7 Trigonometric Functions
L7.4 5 Graphing the Trigonometric
Functions (Part 2)
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The sine function
Imagine a particle on the unit circle, starting
at (1,0) and rotating counterclockwise around the
origin. Every position of the particle
corresponds with an angle, ?, where y sin ?.
As the particle moves through the four quadrants,
we get four pieces of the sin graph
I. From 0 to 90 the y-coordinate
increases from 0 to 1 II. From 90 to 180
the y-coordinate decreases from 1 to 0 III. From
180 to 270 the y-coordinate decreases from 0 to
-1 IV. From 270 to 360 the y-coordinate
increases from -1 to 0
? sin ?
0 0
p/2 1
p 0
3p/2 -1
2p 0
Interactive Sine Unwrap
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Sine is a periodic function p 2p
sin ? Domain (angle measures) all real
numbers, (-8, 8) Range (ratio
of sides) -1 to 1, inclusive -1, 1
sin ? is an odd function it is symmetric wrt the
origin.
sin(-?) -sin(?)
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The cosine function
Imagine a particle on the unit circle, starting
at (1,0) and rotating counterclockwise around the
origin. Every position of the particle
corresponds with an angle, ?, where x cos ?.
As the particle moves through the four quadrants,
we get four pieces of the cos graph
I. From 0 to 90 the x-coordinate
decreases from 1 to 0 II. From 90 to 180
the x-coordinate decreases from 0 to -1 III.
From 180 to 270 the x-coordinate increases from
-1 to 0 IV. From 270 to 360 the x-coordinate
increases from 0 to 1
? cos ?
0 1
p/2 0
p -1
3p/2 0
2p 1
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Cosine is a periodic function p 2p
cos ? Domain (angle measures) all real
numbers, (-8, 8) Range (ratio
of sides) -1 to 1, inclusive -1, 1
cos ? is an even function it is symmetric wrt
the y-axis.
cos(-?) cos(?)
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Tangent Function
Recall that . Since cos ?
is in the denominator, when cos ? 0, tan ? is
undefined. This occurs _at_ p intervals, offset by
p/2 -p/2, p/2, 3p/2, 5p/2, Lets
create an x/y table from ? -p/2 to ? p/2
(one p interval), with 5
input angle values.
? tan ?
-p/2 -8
-p/4 -1
0 0
p/4 1
p/2 8
? sin ? cos ? tan ?
-p/2
-p/4
0
p/4
p/2
0
-8
-1
-1
1
0
0
1
0
1
8
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Graph of Tangent Function Periodic
Vertical asymptotes where cos ? 0
? tan ?
-p/2 -8
-p/4 -1
0 0
p/4 1
p/2 8
-3p/2
3p/2
tan ? Domain (angle measures) ? ? p/2 pn
Range (ratio of sides) all real numbers
(-8, 8)
tan ? is an odd function it is symmetric wrt the
origin.
tan(-?) -tan(?)
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Cotangent Function
Recall that . Since sin ?
is in the denominator, when sin ? 0, cot ? is
undefined. This occurs _at_ p intervals, starting
at 0 -p, 0, p, 2p, Lets create an x/y
table from ? 0 to ? p (one p interval),
with 5 input angle values.
? cot ?
0 8
p/4 1
p/2 0
3p/4 -1
p -8
? sin ? cos ? cot ?
0
p/4
p/2
3p/4
p
1
8
0
1
0
1
0
-1
-8
1
0
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Graph of Cotangent Function Periodic
Vertical asymptotes where sin ? 0
cot ?
? tan ?
0 8
p/4 1
p/2 0
3p/4 -1
p -8
-3p/2
-p
p
-p/2
p/2
3p/2
cot ? Domain (angle measures) ? ? pn
Range (ratio of sides) all real numbers (-8, 8)
cot ? is an odd function it is symmetric wrt the
origin.
tan(-?) -tan(?)
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Cosecant is the reciprocal of sine
Vertical asymptotes where sin ? 0
csc ?
?
0
p
-p
-3p
2p
3p
-2p
sin ?
One period 2p
sin ? and csc ? are odd (symm wrt origin)
csc ? Domain ? ? pn
(where sin ? 0) Range csc ? 1
or (-8, -1 U 1, 8
sin ? Domain (-8, 8) Range -1,
1
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Secant is the reciprocal of cosine
sec ? Domain ? ? p/2 pn
(where cos ? 0) Range sec ?
1 or (-8, -1 U 1, 8
cos ? and sec ? are even (symm wrt y-axis)
cos ? Domain (-8, 8) Range -1,
1
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Summary of Graph Characteristics
Defn ? ? Defn ? ? Period Domain Range Even/Odd
sin ?
csc ?
cos ?
sec ?
tan ?
cot ?
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Summary of Graph Characteristics
Defn ? ? Defn ? ? Period Domain Range Even/Odd
sin ? opp hyp y r 2p (-8, 8) -1 x 1 or -1, 1 odd
csc ? 1 .sin? r .y 2p ? ? pn csc ? 1 or (-8, -1 U 1, 8) odd
cos ? adj hyp x r 2p (-8, 8) All Reals or (-8, 8) even
sec ? 1 . sin? r y 2p ? ? p2 pn sec ? 1 or (-8, -1 U 1, 8) even
tan ? sin? cos? y x p ? ? p2 pn All Reals or (-8, 8) odd
cot ? cos? .sin? x y p ? ? pn All Reals or (-8, 8) odd
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Graphing Trig Functions on the TI89

• Mode critical radian vs. degree
• Graphing ZoomTrig sets x-coordinates as multiple
  of p/2
• Graph the following in radian mode sin(x), cos x
  use trace to observe x/y values
• Switch to degree mode and re-graph the above
• What do you think would happen if you graphed
  cos(x), or 3cos(x) 2? Well study these
  transformations in the next chapter

Enter the function. Use ZoomTrig.                                             This is the graph.