Transcript Angles, Degrees, and Special Triangles

Basic Graphs
Trigonometry
MATH 103
S. Rook
Overview
• Section 4.1 in the textbook:
– The sine graph
– The cosine graph
– The tangent graph
– The cosecant graph
– The secant graph
– The cotangent graph
– Even and odd functions
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The Sine Graph
The Sine Graph
• Recall how to graph a line by
using a table of values:
– e.g. y = 2x
– Pick values for x and then
calculate y for y = sin x
• e.g.
x  0,
  3
,
4 2
,
4
, ,
5 3 7
,
,
,2
4 2 4
– All points on the sine graph can
be written as (x, sin x)
• Obtained from the known
values on the Unit Circle
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The Sine Graph (Continued)
• Finally, plot the
points
• At this point, you
should mainly be
concerned with
recognizing the
shape of the
graphs of the six
trigonometric
functions
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The Sine Graph and the Unit Circle
• Recall that on the
Unit Circle, (x, y)
corresponds to
(cos θ, sin θ)
• For the sine graph, we
are only concerned with
the value of y:
–
–
–
–
QI: y increases to 1
QII: y decreases to 0
QIII: y decreases to -1
QIV: y increases to 0
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Period
• Given a function f(x), the period is the smallest value p such
that f(x + p) = f(x), for all x in f
– i.e. when the graph starts to repeat – the length of ONE
cycle
– All six trigonometric functions are periodic (repeat in an
infinite number of cycles)
• One cycle of y = sin x consists of tracing out the Unit Circle as
θ varies from 0 to 2π
– The graph of y = sin x does not repeat in this interval
• When θ ≥ 2π, the graph of y = sin x starts to repeat
• Therefore, the period of y = sin x is 2π
p = 2π
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Period and Extending Graphs
• To extend the graph of a trigonometric function:
– Utilize its period
– By adding 2π to each x in the previous table of values, we
obtain the next cycle of the sine graph
– By adding 2πk (where k is an integer) to each x in the
previous table of values, we obtain the same y-value in
another cycle of the sine graph
k can be negative meaning the cycles extend infinitely
in both directions
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The Sine Graph (Example)
Ex 1: Extend the table of values for y = sin x to
include the next cycle (i.e. 0 ≤ x ≤ 4π)
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The Cosecant Graph
The Cosecant Graph
• Recall the relationship between
y = sin x and y = csc x:
y  csc x 
1
sin x
• Thus, we take the reciprocal of the
values from the sine graph to
obtain the cosecant graph
• Note that some of the reciprocals
will have a 0 in the denominator:
– Will be undefined (e.g. π)
– The graph will have a vertical
asymptote at these points
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The Cosecant Graph (Continued)
• Plot the points
• Again, be able to
recognize the shape
• Because y = sin x and
y = csc x are
reciprocals, y = csc x
also has a period of
2π
• We can use the same
method to extend the
cosecant graph as we did to extend the sine graph
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The Cosine Graph
The Cosine Graph
• All points on the cosine graph can be
written as (x, cos x)
– Obtained from the known values on
the Unit Circle
• When drawing the cosine graph, we trace
out the entire Unit Circle as θ varies from 0
to 2π without the graph repeating
– Therefore, the period of the cosine
graph is 2π
• By adding 2πk to each x, we get the same
y-value in another cycle of the cosine graph
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The Cosine Graph (Continued)
• Plot the points
• Again, be able to
recognize the
shape of the graph
• We can extend the
cosine graph just
like how like how
we extended the
sine graph
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The Cosine Graph and the Unit Circle
• Recall that on the
Unit Circle, (x, y)
corresponds to
(cos θ, sin θ)
• For the cosine graph, we
are only concerned with
the value of x:
–
–
–
–
QI: x decreases to 0
QII: x decreases to -1
QIII: x increases to 0
QIV: x increases to 1
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The Cosine Graph (Example)
Ex 2: Extend the table of values for y = cos x to
include the previous cycle (i.e. -2π ≤ x ≤ 2π)
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The Secant Graph
The Secant Graph
• Recall the relationship between
y = cos x and y = sec x:
y  sec x 
1
cos x
• Thus, we take the reciprocal of the
values from the cosine graph to
obtain the secant graph
• Note that some of the reciprocals
will have a 0 in the denominator:
– Will be undefined (e.g. π⁄2)
– The graph will have a vertical
asymptote at these points
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The Secant Graph (Continued)
• Plot the points
• Again, be able to
recognize the shape
• Because y = cos x and
y = sec x are
reciprocals, y = sec x
also has a period of
2π
• We can use the same
method to extend the
secant graph as we did to extend the other graphs
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The Tangent Graph
The Tangent Graph
• Recall the relationship of tan x between
sin x and cos x:
sin x
y  tan x 
cos x
• Remember that on the Unit Circle, (x, y)
corresponds to (cos θ, sin θ):
– Thus we can create a table of values
• Again, the graph will have a vertical
asymptote where the tangent is undefined
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The Tangent Graph (Continued)
• Plot the points
• Notice that the period of the
tangent graph is π, NOT 2π
– The graph in QI & QII will be the
SAME as the graph in QIII & QIV
(in reference to the unit circle)
• To extend the tangent graph,
add kπ to each value in the
previous table
– Yields the same y-values in
another cycle
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The Tangent Graph (Example)
Ex 3: Extend the tangent graph from
-π ≤ x ≤ 2π
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The Cotangent Graph
The Cotangent Graph
• Recall the relationship between
y = tan x and y = cot x:
y  cot x 
1
tan x
• Thus, we take the reciprocal of the
values from the tangent graph to
obtain the cotangent graph
• Like the tangent graph, the
cotangent graph has a period of π
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The Cotangent Graph (Continued)
• Plot the points
• As with all the other
graphs, be able to
recognize the shape
• Can extend the graph like
the others
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Even and Odd Functions
Even and Odd Functions
• Recall the definition of even and odd functions:
– If f(-x) = f(x), f(x) is an even function
– If f(-x) = -f(x), f(x) is an odd function
• Examine the Unit Circle at
the right:
cos(-θ) = cos θ meaning?
sin(-θ) = -sin θ meaning?
– Using identities, we
can infer whether the
remaining trigonometric
functions are even or odd
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Even and Odd Functions (Example)
Ex 4: Show or evaluate the following:
a)
 7 
csc 

 4 
b) Show that cot(-θ) is an odd function
c) cos(-θ) tan θ = sin θ
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Summary
• After studying these slides, you should be able
to:
– Recognize the shape of the graphs of the six
trigonometric functions
– State the period for each trigonometric function
– Work with even and odd functions
• Additional Practice
– See the list of suggested problems for 4.1
• Next lesson
– Amplitude, Reflection, and Period (Section 4.2)
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