Transcript Angles, Degrees, and Special Triangles

Graphs of Other Trigonometric
Functions
MATH 109 - Precalculus
S. Rook
Overview
• Section 4.6 in the textbook:
– Graphing the tangent & cotangent functions
– Graphing the secant & cosecant functions
2
Graphing the Tangent &
Cotangent Functions
Graph of the Parent Tangent
Function y = tan x
• Recall that on the unit circle any point (x, y) can be
written as (cos θ, sin θ)
• It can be shown that the period of y = tan x is π
– By examining the unit
circle
• Thus, by taking
y⁄ of each point on
x
the circumference of
the unit circle we
generate one cycle of
y = tan x, 0 < x < π
4
Graph of the Parent Tangent
Function y = tan x (Continued)
• To graph any tangent function we need to know:
– A set of points on the parent function y = tan x
• (0, 0), (π⁄4, 1), (π⁄2 , und), ( 3π⁄4, -1), (π, 0)
– Naturally these are not the only points, but are
often the easiest to manipulate
– The shape of the graph
5
Graph of the Parent Tangent
Function y = cot x
• Recall that on the unit circle any point (x, y) can be
written as (cos θ, sin θ)
• It can be shown that the period of y = cot x is π
– By examining the unit
circle
• Thus, by taking
x⁄ of each point on
y
the circumference of
the unit circle we
generate one cycle of
y = cot x, 0 < x < π
6
Graph of the Parent Cotangent
Function y = cot x (Continued)
• To graph any tangent function we need to know:
– A set of points on the parent function y = tan x
• (0, und), (π⁄4, 1), (π⁄2 , 0), ( 3π⁄4, -1), (π, und)
– Naturally these are not the only points, but are
often the easiest to manipulate
– The shape of the graph
7
Graphing y = d + a tan(bx + c) or
y = d + a cot(bx + c)
• To graph y = d + a tan(bx + c) or y = d + a cot(bx + c) :
– Establish the y-axis
– Establish the x-axis
• The x-values of the 5 points in the are the transformed
x-values for the final graph
– Use transformations to calculate the y-values for the
final graph
– Connect the points in the shape of the tangent or
cotangent – this is 1 cycle
• Be aware of reflection when it exists
– Extend the graph if necessary
8
Graphing the Tangent & Cotangent
Functions
Ex 1: Graph by identifying i) period, phase shift,
and vertical translation ii) extend the graph
one period forwards


a) y  2  tan x  2 
1
b) y   cot 3x 
4
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Graphing the Secant & Cosecant
Functions
Graph of the Parent Secant
Function y = sec x
• Recall that on the unit circle any point (x, y) can be
written as (cos θ, sin θ)
• Period of y = sec x is 2π
– Reciprocal of y = cos x
• Thus, by taking
1⁄ of each point on
x
the circumference of
the unit circle we
generate one cycle of
y = sec x, 0 < x < 2π
11
Graph of the Parent Secant
Function y = sec x (Continued)
• To graph any secant function we need to know:
– A set of points on the parent function y = tan x
• (0, 1), (π⁄2, und), (π, -1), ( 3π⁄2, und), (2π, 1)
– Naturally these are not the only points, but are
often the easiest to manipulate
– The shape of the graph
12
Graph of the Parent Cosecant
Function y = csc x
• Recall that on the unit circle any point (x, y) can be
written as (cos θ, sin θ)
• Period of y = csc x is 2π
– Reciprocal of y = sin x
• Thus, by taking
1⁄ of each point on
y
the circumference of
the unit circle we
generate one cycle of
y = csc x, 0 < x < 2π
13
Graph of the Parent Cosecant
Function y = csc x (Continued)
• To graph any cosecant function we need to know:
– A set of points on the parent function y = csc x
• (0, und), (π⁄4, 1), (π⁄2 , 0), ( 3π⁄4, -1), (π, und)
– Naturally these are not the only points, but are
often the easiest to manipulate
– The shape of the graph
14
Graphing y = d + a sec(bx + c) or
y = d + a csc(bx + c)
• To graph y = d + a sec(bx + c) or y = d + a csc(bx + c) :
– Establish the y-axis
– Establish the x-axis
• The x-values of the 5 points in the are the transformed
x-values for the final graph
– Use transformations to calculate the y-values for the
final graph
– Connect the points in the shape of the secant or
cosecant– this is 1 cycle
• Be aware of reflection when it exists
– Extend the graph if necessary
15
Graphing the Secant & Cosecant
Functions
Ex 2: Graph by identifying i) period, phase shift,
and vertical translation ii) extend the graph
one period backwards
a) y  2  3secx
 x

b) y  2 csc     3
 4

16
Summary
• After studying these slides, you should be able to:
– Graph tangent & cotangent functions
– Graph secant & cosecant functions
• Additional Practice
– See the list of suggested problems for 4.6
• Next lesson
– Inverse Trigonometric Functions (Section 4.7)
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