Summary Ch5 1_5 4 com

8/8/2019 Summary Ch5 1_5 4 com

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Trigonometry

Summary Ch 5.1-5.4


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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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Trigonometry Topics

Radian Measure

The Unit Circle

Trigonometric Functions

Larger Angles

Graphs of the Trig Functions

Trigonometric IdentitiesSolving Trig Equations


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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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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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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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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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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 .



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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.


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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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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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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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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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.


2


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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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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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Cofunction Identities Thefunctionof an angle = the cofunction of its

complement.


sin cos( )x x= 90

sec csc( )x x= 90

tan cot( )x x= 90


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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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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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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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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

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