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8/8/2019 Summary Ch5 1_5 4 com
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Trigonometry
Summary Ch 5.1-5.4
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Carolyn
C.Wheate r,
2000
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Trigonometry
Trigonometry begins in the rightTrigonometry begins in the right
triangle, but it doesnt have to betriangle, but it doesnt have to berestricted to triangles. Therestricted to triangles. The
trigonometric functions carry thetrigonometric functions carry the
ideas of triangle trigonometry into aideas of triangle trigonometry into a
broader world of real-valuedbroader world of real-valuedfunctions and wave forms.functions and wave forms.
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C.Wheate r,
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Trigonometry Topics
Radian Measure
The Unit Circle
Trigonometric Functions
Larger Angles
Graphs of the Trig Functions
Trigonometric IdentitiesSolving Trig Equations
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C.Wheate r,
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Radian Measure
To talk about trigonometric functions, it is
helpful to move to a different system of
angle measure, called radian measure.A radian is the measure of a central angle
whose interceptedarc is equal in length to
the radius of the circle.
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Radian Measure
degrees
360
radians
=
2
There are 2 radians in a full rotation --
once around the circle
There are 360 in a full rotation
To convert from degrees to radians or
radians to degrees, use the proportion
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Sample Problems
Find the degree
measure equivalent
of radians.degrees
360
radians
210360
r
=
=
=
= =
2
2
360 420
420
360
7
6
r
r
degrees
360
radians
3603 4
=
=
=
=
2
2
2 270
135
d
d
d
3
4
Find the radian
measure equivalent
of 210
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The Unit Circle
Imagine a circle on the
coordinate plane, with its
center at the origin, anda radius of 1.
Choose a point on the
circle somewhere inquadrant I.
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The Unit Circle
Connect the origin to the
point, and from that point
drop a perpendicular tothe x-axis.
This creates a right
triangle with hypotenuseof 1.
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C.Wheate r,
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The Unit Circle
sin( ) = =y
y1
cos bg= =x x1
x
y1
is the
angle of
rotation
The length of its legs are
the x- and y-coordinates of
the chosen point.Applying the definitions of
the trigonometric ratios to
this triangle gives
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The Unit Circle
sin( ) = =y
y
1
cos bg= =x x1
The coordinates of the chosen point are the
cosine and sine of the angle .
This provides a way to define functions sin( )and cos( ) for all real numbers .
The other trigonometric functions can be
defined from these.
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Trigonometric Functions
sin( ) = y
cos bg= x
tan bg= yx
csc bg= 1y
sec bg= 1x
cot bg= xy
x
y1
is the
angle of
rotation
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Around the Circle
As that point moves
around the unit
circle intoquadrants II, III,
and IV, the new
definitions of the
trigonometricfunctions still hold.
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C.Wheate r,
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Reference Angles
The angles whose terminal sides fall in
quadrants II, III, and IV will have values of
sine, cosine and other trig functions whichare identical (except for sign) to the values
of angles in quadrant I.
The acute angle which produces the samevalues is called the reference angle.
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Reference Angles
The reference angle is the angle between
the terminal side and the nearest arm of the
x-axis.The reference angle is the angle, with vertex
at the origin, in the right triangle created by
dropping a perpendicular from the point onthe unit circle to the x-axis.
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Quadrant II
Original angle
Reference angle
For an angle, , in
quadrant II, the
reference angle is
In quadrant II,
sin( ) is positivecos( ) is negative
tan( ) is negative
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C.Wheate r,
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Quadrant III
Original angle
Reference angle
For an angle, , in
quadrant III, the
reference angle is -
In quadrant III,
sin( ) is negativecos( ) is negative
tan( ) is positive
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Quadrant IV
Original angle
Reference angle
For an angle, , in
quadrant IV, the
reference angle is2
In quadrant IV,
sin( ) is negativecos( ) is positive
tan( ) is negative
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All Star Trig Class
Use the phrase All Star Trig Class to
remember the signs of the trig functions in
different quadrants.AllStar
Trig Class
All functions
are positiveSine is positive
Tan is positive Cos is positive
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C.Wheate r
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SineThe most fundamental sine wave, y=sin(x),
has the graph shown.It fluctuates from 0 to a high of 1, down to 1,
and back to 0, in a space of 2 .
Graphs of the Trig Functions
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C.Wheate r
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The graph of is determined
by four numbers, a, b, h, and k.
The amplitude, a, tells the height of each peak andthe depth of each trough.
The frequency, b, tells the number of full wave
patterns that are completed in a space of 2 .
The period of the function is
The two remaining numbers, h and k, tell the
translation of the wave from the origin.
Graphs of the Trig Functions
y a b x h k = +sin b gc h
2
b
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Sample Problem
Which of the following
equations best describes
the graph shown? (A)y = 3sin(2x) - 1(B)y = 2sin(4x)
(C)y = 2sin(2x) - 1
(D)y = 4sin(2x) - 1
(E)y = 3sin(4x)
2 1 1 2
5
4
3
2
1
1
2
3
4
5
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C.Wheate r
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Sample Problem
Find the baseline between the
high and low points.
Graph is translated -1vertically.
Find height of each peak.
Amplitude is 3
Count number of waves in 2Frequency is 2
2 1 1 2
5
4
3
2
1
1
2
3
4
5
y = 3sin(2x) - 1
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C.Wheate r
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Graphs of the Trig Functions
y a b x h k = +cos b gc hAmplitude a Height of each peak
Frequency b Number of full wave patternsPeriod 2 /b Space required to complete waveTranslation h, k Horizontal and vertical shift
The values ofa, b, h, and kchange the shape
and location of the wave as for the sine.
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C.Wheate r
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Which of the following
equations best describes
the graph? (A)y = 3cos(5x) + 4(B)y = 3cos(4x) + 5
(C)y = 4cos(3x) + 5(D)y = 5cos(3x) +4
(E)y = 5sin(4x) +3
Sample Problem
2 1 1 2
8
6
4
2
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C.Wheate r
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Find the baseline Vertical translation + 4
Find the height of peak Amplitude = 5
Number of waves in
2Frequency =3
Sample Problem
2 1 1 2
8
6
4
2
y = 5cos(3x) + 4
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Tangent The tangent function has a
discontinuous graph,repeating in a period of .
Cotangent
Like the tangent, cotangent isdiscontinuous.
Discontinuities of the cotangent
are units left of those for
tangent.
Graphs of the Trig Functions
2
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C.Wheate r
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Graphs of the Trig Functions
y=sec(x)
Secant and Cosecant The secant and cosecant functions are the
reciprocals of the cosine and sine functionsrespectively.
Imagine each graph is balancing on the peaks and
troughs of its reciprocal function.
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Trigonometric Identities
An identity is an equation which is true for
all values of the variable.
There are many trig identities that are usefulin changing the appearance of an
expression.
The most important ones should becommitted to memory.
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Trigonometric Identities
Reciprocal Identities
sin cscx x=
1
cossec
xx
=1
tancot
xx
=1
tan sincos
x xx
=
cot
cos
sinx
x
x=
Quotient Identities
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C.Wheate r
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Cofunction Identities Thefunctionof an angle = the cofunction of its
complement.
Trigonometric Identities
sin cos( )x x= 90
sec csc( )x x= 90
tan cot( )x x= 90
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Trigonometric Identities
sin cos2 2 1x x+ =
1 2 2+ =cot cscx x
tan sec2 2
1x x+ =
Pythagorean Identities
The fundamental Pythagorean identity
Divide the first by sin2x Divide the first by cos2x
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Solving Trig Equations
Solve trigonometric equations by following
these steps:
If there is more than one trig function, useidentities to simplify
Let a variable represent the remaining function
Solve the equation for this new variableReinsert the trig function
Determine the argument which will produce the
desired value
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Solving Trig Equations
To solving trig equations:
Use identities to simplifyLet variable = trig function
Solve for new variable
Reinsert the trig function
Determine the argument
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C.Wheate r
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SolveUse the Pythagorean
identity
(cos2x = 1 - sin2x)
Distribute
Combine like terms
Order terms
Sample Problem
3 3 2 0
3 3 2 1 0
3 3 2 2 0
1 3 2 02 3 1 0
2
2
2
2
2
=
=
+ =
+ =
+ =
sin cos
sin sin
sin sin
sin sinsin sin
x x
x x
x x
x xx x
c h
3 3 2 02 =sin cosx x
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C.Wheate r
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Lett= sinx
Factor and solve.
Sample Problem
Solve 3 3 2 02 =sin cosx x
2 3 1 02sin sinx x + =
2 3 1 02 1 1 0
2 1 0 1 0
2 1 11
2
2
t t
t t
t t
t t
t
+ =
=
= =
= =
=
( )( )
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C.Wheate r
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Sample Problem
Solve 3 3 2 02
=sin cosx x
x =
6
5
6or
x =
2
x =
6
5
6 2, ,
Replacet= sinx.
t=sin(x) = when
t=sin(x) = 1 when
So the solutions are