trigonometry handbook (formula)

8/11/2019 Trigonometry Handbook (Formula)

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Copyright20122014,EarlWhitney,RenoNV. AllRightsReserved

MathHandbook

ofFormulas,ProcessesandTricks

Trigonometry

Preparedby: EarlL.Whitney,FSA,MAAA

Version

1.05

September

12,

2014


2/46

TrigonometryHandbook

This

is

an

early

work

product

that

will

eventually

result

in

an

extensive

handbook

on

the

subject

ofTrigonometry. Initscurrentform,thehandbookcoversmanyofthesubjectscontainedina

Trigonometrycourse,butisnotexhaustive. Inthemeantime,wearehopefulthatthismaterial

willbehelpfultothestudent. Revisionstothishandbookwillbeprovidedonwww.mathguy.us

astheybecomeavailable.

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Page Description

Chapter1:FunctionsandSpecialAngles

6 Definitions(x andy axes)

6 Radians

6 SineCosineRelationship

7 Definitions(RightTriangle)

7 SOHCAHTOA

7 TrigFunctionsofSpecialAngles

8 TrigFunctionValuesinQuadrantsII,III,andIV

9 TheUnitCircle

Chapter

2:

Graphs

of

Trig

Functions10 BasicTrigFunctions

11 TableofTrigFunctionCharacteristics

12 SineFunction

14 CosineFunction

16 TangentFunction

18 CotangentFunction

20 SecantFunction

22 CosecantFunction

Chapter3:InverseTrigonometricFunctions

24 Definitions24 PrincipalValues

25 GraphsofInverseTrigFunctions

Chapter4:KeyAngleFormulas

26 AngleAdditionFormulas

26 DoubleAngleFormulas

26 Half AngleFormulas

27 PowerReducingFormulas

27 ProducttoSumFormulas

27 SumtoProductFormulas

28 Cofunctions28 LawofSines

28 LawofCosines

28 PythagoreanIdentities


TableofContents

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TableofContents

Page Description

Chapter5:SolvinganObliqueTriangle

29 SummaryofMethods

30 TheAmbiguousCase

32 FlowchartfortheAmbiguousCase

Chapter6:AreaofaTriangle

33 GeometryFormula

33 Heron'sFormula

34 TrigonometricFormulas

34 CoordinateGeometryFormula

Chapter7:PolarCoordinates

35 Introduction

35 ConversionbetweenRectangularandPolarCoordinates

36 ExpressingComplexNumbersinPolarForm

36 OperationsonComplexNumbersinPolarForm

37 DeMoivre'sTheorem

38 DeMoivre'sTheoremforRoots

Chapter8:GraphingPolarFunctions

39 Cardioid

40 Rose

Chapter9:Vectors

41 Introduction

41 SpecialUnitVectors

41 VectorComponents

42 VectorProperties

43 DotProduct

44 VectorProjection

44 OrthogonalComponentsofaVector

44 Work

45 Index

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TableofContents

UsefulWebsites

http://www.mathguy.us/

http://mathworld.wolfram.com/

www.khanacademy.org

http://www.analyzemath.com/Trigonometry.html

SchaumsOutline

Note: This study guide was prepared to be a companion to most books on the subject of High

School Trigonometry. Precalculus (4th edition) by Robert Blitzer was used to determine some of the

subjects to include in this guide.

Animportantstudentresourceforanyhighschoolorcollegemathstudentis

aSchaumsOutline. Eachbookinthisseriesprovidesexplanationsofthe

varioustopicsinthecourseandasubstantialnumberofproblemsforthe

studenttotry. Manyoftheproblemsareworkedoutinthebook,sothe

studentcanseeexamplesofhowtheyshouldbesolved.

SchaumsOutlinesareavailableatAmazon.com,Barnes&Nobleandother

booksellers.

Mathguy.us

Developed

specifically

for

math

students

from

Middle

School

to

College,

based

on

the

author'sextensiveexperienceinprofessionalmathematicsinabusinesssettingandinmathtutoring.

Containsfreedownloadablehandbooks,PCApps,sampletests,andmore.

WolframMathWorldPerhapsthepremiersiteformathematicsontheWeb. Thissitecontains

definitions,explanationsandexamplesforelementaryandadvancedmathtopics.

KhanAcademySuppliesafreeonlinecollectionofthousandsofmicrolecturesviaYouTubeon

numeroustopics.

It's

math

and

science

libraries

are

extensive.

AnalyzeMathTrigonometryContainsfreeTrigonometrytutorialsandproblems. UsesJavaappletsto

exploreimportanttopicsinteractively.

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Trigonometry

Trigonometric Functions

Trigonometric Functions(- and - axes)

Radians ( =)

0 = 0 radians

30 =6

radians

45 =4

radians

60 =

3 radians

90 =2

radians

sin = sin =1

csc

cos = cos =1

sec

tan = tan =1

cot tan =sin cos

cot = cot =

1

tan cot =cos sin

sec = sec =1

cos

csc = csc =1

sin

sin + 2 = cos

sin = cos 2

sin2 + cos2 = 1

Sine-Cosine Relationship

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Trigonometry

Trigonometric Functions and Special Angles

Trigonometric Functions (Right Triangle)

Special Angles

Trig Functions of Special Angles ()

Radians Degrees

0 0 =

=

=

30 =

=

45

=

60 =

=

90 = =

undefined

SOH-CAH-TOA

sin =

sin = sin =

cos =

cos = cos =

tan = tan = tan =

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Trigonometry

Trigonometric Function Values in Quadrants II, III, and IV

In quadrants other than Quadrant I, trigonometric values for angles are calculated in the following

manner:

Draw the angle on the Cartesian Plane.

Calculate the measure of the angle from the x-

axis to .

Find the value of the trigonometric function of

the angle in the previous step.

Assign a + or sign to the trigonometric

value based on the function used and the

quadrant is in.

Examples:

in Quadrant II Calculate: (180 )For = 120, base your work on 180 120 = 60sin 60 =

32 , so: =

in Quadrant III Calculate: ( 180)For = 210, base your work on 210 180 = 30cos 30 =

32 , so: =

in Quadrant IV Calculate: (360 )For = 315, base your work on 360 315 = 45tan45 =1, so: =

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Trigonometry

The Unit Circle

The Unit Circle diagram below provides

- and

-values on a circle of radius

1at key angles. At any

point on the unit circle, the -coordinate is equal to the cosine of the angle and the -coordinate isequal to the sine of the angle. Using this diagram, it is easy to identify the sines and cosines of angles

that recur frequently in the study of Trigonometry.

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The sine and cosecant functions are reciprocals. So:

sin

=

1

csc and csc

=

1

sin

The cosine and secant functions are reciprocals. So:

cos = 1sec and sec =

1

cos

The tangent and cotangent functions are reciprocals. So:

tan = 1cot and cot =

1

tan

Trigonometry

Graphs of Basic (Parent) Trigonometric Functions

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Trigonometry

SummaryofCharacteristicsandKeyPointsTrigonometricFunctionGrap

Function: Sine Cosine Tangent Cotangent

ParentFunction sin cos tan cot

Domain , , , except

,

whereisodd

, except,

whereisanInteger

wh

VerticalAsymptotes none none

,whereisodd

,whereisan

Integer

Range 1,1 1,1 , ,

Period 2 2

intercepts ,whereisanInteger

,whereisodd

midwaybetween

asymptotes

midwaybetween

asymptotes

Oddor

Even

Function

(1)

Odd

Function

Even

Function

Odd

Function

Odd

Function

Ev

GeneralForm sin cos tan cot s

Amplitude,Period,

PhaseShift,VerticalShift||,

2

,

, ||,

2

,

, ||,

,

, ||,

,

, |

when (2)

verticalasymptote

when

vert

when

verticalasymptote

when

vert

when verticalasymptote

Notes:

(1) Anoddfunctionissymmetricabouttheorigin,i.e. . Anevenfunctionissymmetricabouttheaxis,i.e.,

(2) AllPhaseShiftsaredefinedtooccurrelativetoastartingpointoftheaxis(i.e.,theverticalline 0).

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Trigonometry

Graph of a General Sine Function

General Form

The general form of a sine function is: = () + .In this equation, we find several parameters of the function which will help us graph it. In particular:

Amplitude: = ||. The amplitude is the magnitude of the stretch or compression of thefunction from its parent function: = sin.

Period: = . The period of a trigonometric function is the horizontal distance over whichthe curve travels before it begins to repeat itself (i.e., begins a new cycle). For a sine or cosine

function, this is the length of one complete wave; it can be measured from peak to peak or

from trough to trough. Note that 2is the period of = sin . Phase Shift: =. The phase shift is the distance of the horizontal translation of the

function. Note that the value of in the general form has a minus sign in front of it, just like does in the vertex form of a quadratic equation: = ( ) + . So,

o A minus sign in front of the implies a translation to the right, ando A plus sign in front of the implies a implies a translation to the left.

Vertical Shift: =. This is the distance of the vertical translation of the function. This isequivalent to

in the vertex form of a quadratic equation:

= (

) +

.

Example: = + has the equationThe midline y = D In this example, the midline.

is: y = 3. One wave, shifted to the right, is shown in orange below.

=; =; = ; =For this example:

Amplitude: = || = || = Period: = = =Phase Shift: = =

=

Vertical Shift: = =

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Trigonometry

Graphing a Sine Function with No Vertical Shift: = ()

Step 1: Phase Shift: =.The first wave begins at the

point units to the right ofthe Origin.

= = =

.

The point is: ,

Step 2: Period: = .The first wave ends at the

point

units to the right of

where the wave begins. + , =,

= = =. The firstwave ends at the point:

Step 3: The third zero point

is located halfway between

the first two.

The point is:

+ ,=,

Step 4: The -value of thepoint halfway between the

left and center zero points is

"".

The point is:

+

,= (,)


center and right zero points

is .

The point is:

+ ,=,

Step 7: Duplicate the wave

to the left and right as

desired.

Step 6: Draw a smooth

curve through the five keypoints.

This will produce the graphof one wave of the function.

Example:

=

.

A wave (cycle) ofthe sine functionhas three zero points (points on the x-axis)

at the beginning of the period, at the end of the period, and halfway in-between.

Note: If 0, all pointson the curve are shifted

vertically by units.

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Trigonometry

Graph of a General Cosine Function

General Form

The general form of a cosine function is: = () + .In this equation, we find several parameters of the function which will help us graph it. In particular:

Amplitude: = ||. The amplitude is the magnitude of the stretch or compression of thefunction from its parent function: = cos .

Period: = . The period of a trigonometric function is the horizontal distance over whichthe curve travels before it begins to repeat itself (i.e., begins a new cycle). For a sine or cosine

function, this is the length of one complete wave; it can be measured from peak to peak or

from trough to trough. Note that 2is the period of = cos . Phase Shift: =. The phase shift is the distance of the horizontal translation of the





= (

) +

.


is: y = 3. One wave, shifted to the right, is shown in orange below.

=; =; = ; =For this example:


=

Vertical Shift: = =

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Trigonometry

Graphing a Cosine Function with No Vertical Shift: =()

Step 1: Phase Shift: =.The first wave begins at the

point units to the right ofthe point (,).

= = =

, =

The point is: ,

Step 2: Period: = .The first wave ends at the

point units to the right ofwhere the wave begins. + , =,

= = =. The firstwave ends at the point:

Step 3: The -value of thepoint halfway between those

in the two steps above is

" ".

The point is:

+ ,=,


left and center extrema is

"".

The point is:

+ ,= (,)


center and right extrema is

"".

The point is:

+ ,=,



desired.


curve through the five key

points.

This will produce the graph

of one wave of the function.

Example:

= .

A wave (cycle) ofthe cosine functionhas two maxima (or minima if < 0) one at the beginning of the period and one at the end of the period and a

minimum (or maximum if

< 0) halfway in-between.



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For this example:


=

Vertical Shift: = =

=; =; = ; =

Trigonometry

Graph of a General Tangent Function

General Form

The general form of a tangent function is: = () + .In this equation, we find several parameters of the function which will help us graph it. In particular:

Amplitude: = ||. The amplitude is the magnitude of the stretch or compression of thefunction from its parent function: = tan .

Period: =. The period of a trigonometric function is the horizontal distance over whichthe curve travels before it begins to repeat itself (i.e., begins a new cycle). For a tangent or

cotangent function, this is the horizontal distance between consecutive asymptotes (it is also

the distance between -intercepts). Note that is the period of = tan . Phase Shift: =. The phase shift is the distance of the horizontal translation of the





= (

) +

.


is: y = 3. One cycle, shifted to the right, is shown in orange below.

Note that, for the

tangent curve, we

typically graph half

of the principal

cycle at the point

of the phase shift,

and then fill in the

other half of the

cycle to the left

(see next page).

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Trigonometry

Graphing a Tangent Function with No Vertical Shift: = ()

Step 1: Phase Shift: =.The first cycle begins at the

zero point units to theright of the Origin.

= = =

.

The point is: ,

Step 2: Period: =.Place a vertical asymptote

units to the right of the

beginning of the cycle.

= = =. =.The right asymptote is at:

= + =Step 3: Place a vertical

asymptote units to the

left of the beginning of the

cycle.

The left asymptote is at:

= =


zero point and the right

asymptote is "".

The point is:

+

,=,


left asymptote and the zero

point is " ".

The point is:

+ ,=,



desired.


curve through the three keypoints, approaching the

asymptotes on each side.


Example:

=

.

A cycle ofthe tangent functionhas two asymptotes and a zero point halfway in-

between. It flows upward to the right if

> 0 and downward to the right if

< 0.



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For this example:


=

Vertical Shift: = =

=; =; = ; =

Trigonometry

Graph of a General Cotangent Function

General Form

The general form of a cotangent function is: = () + .In this equation, we find several parameters of the function which will help us graph it. In particular:

Amplitude: = ||. The amplitude is the magnitude of the stretch or compression of thefunction from its parent function: = cot.

Period: =. The period of a trigonometric function is the horizontal distance over whichthe curve travels before it begins to repeat itself (i.e., begins a new cycle). For a tangent or

cotangent function, this is the horizontal distance between consecutive asymptotes (it is also

the distance between -intercepts). Note that is the period of = cot. Phase Shift: =. The phase shift is the distance of the horizontal translation of the





= (

) +

.



Note that, for the

cotangent curve,

we typically graph

the asymptotes

first, and then

graph the curve

between them (see

next page).

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Trigonometry

Graphing a Cotangent Function with No Vertical Shift: =()

= = =.The right asymptote is at:

=

+

=5

Step 1: Phase Shift: =.Place a vertical asymptoteunits to the right of the-axis.

= = =

. The left

asymptote is at: =

Step 2: Period: =.Place another vertical

asymptote units to theright of the first one.

Step 3: A zero point exists

halfway between the two

asymptotes.

The point is: + ,= (,)


left asymptote and the zero

point is "".

The point is:

+

,=,


zero point and the right

asymptote is " ".

The point is:

+ ,=,



desired.


curve through the three key

points, approaching the

asymptotes on each side.


Example:

=

.

A cycle ofthe cotangent functionhas two asymptotes and a zero point halfway in-

between. It flows downward to the right if

> 0 and upward to the right if

< 0.



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For this example:


=

Vertical Shift: = =

=; =; = ; =

Trigonometry

Graph of a General Secant Function

General Form

The general form of a secant function is: = () + .In this equation, we find several parameters of the function which will help us graph it. In particular:

Amplitude: = ||. The amplitude is the magnitude of the stretch or compression of thefunction from its parent function: = sec.

Period: = . The period of a trigonometric function is the horizontal distance over whichthe curve travels before it begins to repeat itself (i.e., begins a new cycle). For a secant or

cosecant function, this is the horizontal distance between consecutive maxima or minima (it is

also the distance between every second asymptote). Note that 2is the period of = sec. Phase Shift: =. The phase shift is the distance of the horizontal translation of the





= (

) +

.



One cycle of the secant curve contains two U-shaped curves, one

opening up and one opening down.

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Trigonometry

Graphing a Secant Function with No Vertical Shift: = ()

Step 1: Graph one wave of

the corresponding cosine

function.

= ( )

The equation of the

corresponding cosine

function for the example is:

=

Step 2: Asymptotes for the

secant function occur at the

zero points of the cosine

function.

The zero points occur at:

(, 0) and , 0Secant asymptotes are:

= and =

Step 3: Each maximum of

the cosine function

represents a minimum for

the secant function.

Cosine maxima and,

therefore, secant minima are

at: , 4 and , 4

Step 4: Each minimum of

the cosine function

represents a maximum for

the secant function.

The cosine minimum and,

therefore, the secant

maximum is at: 5,4

Step 5: Draw smooth U-

shaped curves through each

key point, approaching theasymptotes on each side.



desired. Erase the cosine

function if necessary.



Example:

= .

A cycle ofthe secant functioncan be developed by first plotting a cycle of the

corresponding cosine function because sec

=

cos.

The cosine functions zero points produce asymptotes for the secant function.

Maxima for the cosine function produce minima for the secant function.

Minima for the cosine function produce maxima for the secant function.

Secant curves are U-shaped, alternately opening up and opening down.



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For this example:


=

Vertical Shift: = =

=; =; = ; =

Trigonometry

Graph of a General Cosecant Function

General Form

The general form of a cosecant function is: = () + .In this equation, we find several parameters of the function which will help us graph it. In particular:

Amplitude: = ||. The amplitude is the magnitude of the stretch or compression of thefunction from its parent function: = csc.

Period: = . The period of a trigonometric function is the horizontal distance over whichthe curve travels before it begins to repeat itself (i.e., begins a new cycle). For a secant or

cosecant function, this is the horizontal distance between consecutive maxima or minima (it is

also the distance between every second asymptote). Note that 2is the period of = csc. Phase Shift: =. The phase shift is the distance of the horizontal translation of the





= (

) +

.



One cycle of the cosecant curve contains two U-shaped curves, one

opening up and one opening down.

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Trigonometry

Graphing a Cosecant Function with No Vertical Shift: = ()

Step 1: Graph one wave of

the corresponding sine

function.

= ( )

The equation of the

corresponding sine function

for the example is:

=

Step 2: Asymptotes for the

cosecant function occur at

the zero points of the sine

function.

The zero points occur at:

, 0 , 5, 0 , , 0Cosecant asymptotes are:

=, =5, =Step 3: Each maximum of

the sine function represents

a minimum for the cosecant

function.

The sine maximum and,

therefore, the cosecant

minimum is at: (, 4)

Step 4: Each minimum of

the sine function represents

a maximum for the cosecant

function.

The sine minimum and,

therefore, the cosecant

maximum is at: ,4

Step 5: Draw smooth U-

shaped curves through each

key point, approaching theasymptotes on each side.



desired. Erase the sine

function if necessary.



Example:

= .

A cycle ofthe cosecant functioncan be developed by first plotting a cycle of the

corresponding sine function because csc

=

sin.

The sine functions zero points produce asymptotes for the cosecant function.

Maxima for the sine function produce minima for the cosecant function.

Minima for the sine function produce maxima for the cosecant function.

Cosecant curves are U-shaped, alternately opening up and opening down.



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Trigonometry

Inverse Trigonometric Functions

Inverse Trigonometric Functions

Inverse trigonometric functions ask the question: which angle has a function value of? For example: = sin(0.5) asks which angle has a sine value of 0.5. It is equivalent to: sin = 0.5. = tan 1 asks which angle has a tangent value of 1. It is equivalent to: tan = 1.

Principal Values of Inverse Trigonometric Functions

There are an infinite number of angles that answer these questions.

So, mathematicians have defined a principal solution for problems

involving inverse trigonometric functions. The angle which is the

principal solution (or principal value) is defined to bethe solution that

lies in the quadrants identified in the figure at right. For example:

The solutions to = sin 0.5 are (6 + 2) (56 +2). That is, the set of all solutions to this equation contains thetwo solutions in the interval [0,2), as well as all angles that areinteger multiples of 2less or greater than those two angles.Given the confusion this can create, mathematicians defined a

principal value for the solution to these kinds of equations.

The principal value of for which = sin 0.5lies in Q1 because 0.5 is positive, and is =6.

Ranges of Inverse Trigonometric Functions

The ranges of the inverse trigonometric

functions are the ranges of the principal values

of those functions. A table summarizing these

is provided in the table at right.

Angles in Q4 are generally expressed as

negative angles.

Ranges of Inverse Trigonometric Functions

Function Range

sin 2 2cos 0 tan

2

2

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Trigonometry

Graphs of Inverse Trigonometric Functions

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Trigonometry

Key Angle Formulas

Angle Addition Formulas

sin ( + ) = sin cos + cos sin cos ( + ) = cos cos sin sinsin ( ) = sin cos cos sin cos ( ) = cos cos + sin sin

tan ( + ) = tan+tantan tan tan ( ) = tantan+tan tan

Double Angle Formulas

sin 2 = 2 sin cos cos 2 = cos sin = 1 2sin = 2cos 1

tan 2 = tantan

Half Angle Formulas

sin = cos

cos = +cos

tan

=

cos+cos

=cossin

=sin

+cos

The use of a + or - sign in the half angle

formulas depends on the quadrant in which

the angleresides. See chart below.

Signs of Trig Functions

By Quadrant

sin + sin +

cos - cos +

tan - tan +

sin - sin -

cos - cos +

tan + tan

y

x

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Trigonometry

Key Angle Formulas (contd)

Power Reducing Formulas

sin = cos cos = +cos tan = cos+cos

Product-to-Sum Formulas

= [( ) ( + )] = [( ) + ( + )] = [( + ) + ( )]

=

[

(

+

)

(

)]

Sum-to-Product Formulas

+ = + =

+

+ = + = +

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Trigonometry

Key Angle Formulas (contd)

Cofunctions

Each trigonometric function has a cofunction with symmetric properties in Quadrant I. The following

identities express the relationships between cofunctions.

sin = cos(90 ) cos = sin(90 )tan = cot(90 ) cot = tan(90 )sec = csc(90 ) csc = sec(90 )

Law of Sines (see above illustration)

sin =

sin =

sin

Pythagorean Identities (for any angle )

sin + cos = 1sec = 1 + tan csc = 1 + cot

C

c b

a

A

B

Law of Cosines (see above illustration)

= + 2 cos = + 2 cos = + 2 cos

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Trigonometry

Solving an Oblique Triangle

Several methods exist to solve an oblique triangle, i.e., a triangle with no right angle. The appropriate

method depends on the information available for the triangle. All methods require that the length ofat least one side be provided. In addition, one or two angle measures may be provided. Note that if

two angle measures are provided, the measure of the third is determined (because the sum of all

three angle measures must be 180). The methods used for each situation are summarized below.Given Three Sides and no Angles (SSS)

Given three segment lengths and no angle measures, do the following:

Use the Law of Cosines to determine the measure of one angle.

Use the Law of Sines to determine the measure of one of the two remaining angles.

Subtract the sum of the measures of the two known angles from 180to obtain the measureof the remaining angle.

Given Two Sides and the Angle between Them (SAS)

Given two segment lengths and the measure of the angle that is between them, do the following:

Use the Law of Cosines to determine the length of the remaining leg.

Use the Law of Sines to determine the measure of one of the two remaining angles.

Subtract the sum of the measures of the two known angles from 180to obtain the measureof the remaining angle.

Given One Side and Two Angles (ASA or AAS)

Given one segment length and the measures of two angles, do the following:

Subtract the sum of the measures of the two known angles from 180to obtain the measure

of the remaining angle.

Use the Law of Sines to determine the lengths of the two remaining legs.

Given Two Sides and an Angle not between Them (SSA)

This is the Ambiguous Case. Several possibilities exist, depending on the lengths of the sides and the

measure of the angle. The possibilities are discussed on the next several pages.

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Trigonometry

Solving an Oblique Triangle (contd)

The Ambiguous Case (SSA)

Given two segment lengths and an angle that is not between them, it is not clear whether a triangle is

defined. It is possible that the given information will define a single triangle, two triangles, or even no

triangle. Because there are multiple possibilities in this situation, it is called the ambiguous case.

Here are the possibilities:

There are three cases in which


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Trigonometry

Solving a Triangle (contd)

Solving the Ambiguous Case (SSA)

How do you solve the triangle in each of the cases discussed above. Assume the information given is

the lengths of sides and , and the measure of Angle. Use the following steps:Step 1: Calculate the sine of the missing angle (in this development, angle ).Step 2: Consider the value of sin:

If sin > 1, then we have Case 1 there is no triangle. Stop here.

If sin = 1, then = 90, and we have Case 2 a right triangle. Proceed to Step 4.

If sin < 1, then we have Case 3 or Case 4. Proceed to the next step to determine which.Step 3: Consider whether >.

If


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Trigonometry

Solving an Oblique Triangle (contd)

Solving the Ambiguous Case (SSA) contd

Step 4: Calculate . At this point, we have the lengths of sides and , and the measures of Anglesand . If we are dealing with Case 3 two triangles, we must perform Steps 4 and 5 for each angle.Step 4 is to calculate the measure of Angle as follows: = 180 Step 5: Calculate . Finally, we calculate the value of using the Law of Sines. Note that in the casewhere there are two triangles, there is an Angle in each. So, the Law of Sines should be usedrelating Angles

and

.

sin = sin = sinsin

Ambiguous Case Flowchart

no

yes

Two triangles

Calculate , and then .Steps 4 and 5, above

Start Here

> 1

= 1

< 1Is >?sinValue of

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Trigonometry

Area of a Triangle

Area of a TriangleThere are two formulas for the area of a triangle, depending on what information about the triangle

is available.

Formula 1: The formula most familiar to the student can be used when the base and height of the

triangle are either known or can be determined.

=

where, is the length of the base of the triangle.is the height of the triangle.Note: The base can be any side of the triangle. The height is the measure of the altitude of

whichever side is selected as the base. So, you can use:

or or

Formula 2: Herons formulafor the area of a triangle can be used when

the lengths of all of the sides are known. Sometimes this formula, though

less appealing, can be very useful.

=

(

)(

)(

)

where, = = ( + + ). Note: is sometimes called the semi-perimeter of the triangle.,, are the lengths of the sides of the triangle.

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Trigonometry

Area of a Triangle (contd)

Trigonometric Formulas

The following formulas for the area of a triangle come from trigonometry. Which one is used

depends on the information available:

Two angles and a side:

= =

=

Two sides and an angle:

= = =

Coordinate Geometry

If the three vertices of a triangle are displayed in a coordinate plane, the formula below, using a

determinant, will give the area of a triangle.

Let the three points in the coordinate plane be: (,), (,), (,). Then, the area of thetriangle is one half of the absolute value of the determinant below:

=

Example: For the triangle in the figure at right, the area is:

= =

+ = =

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Trigonometry

Polar Coordinates

Polar coordinates are an alternative method of describing a point in a Cartesian plane based on the

distance of the point from the origin and the angle whose terminal side contains the point. First, letsinvestigate the relationship between a points rectangular coordinates (,)and its polarcoordinates (,).The magnitude, , is the distance of the point from the

origin: = + The angle, , is the angle the line from the point to the

origin makes with the positive portion of the x-axis.

Generally, this angle is expressed in radians, not degrees.

tan = or = tan Conversion from polar coordinates to rectangular coordinates is straightforward:

= cos and = sinExample 1: Express the rectangular form (-4, 4) in polar coordinates:

Given: =4 = 4

=

+

=

(

4) + 4 = 4

2

= tan = tan = tan(1) in Quadrant II, so = So, the coordinates of the point are as follows:

Rectangular coordinates: (4, 4) Polar Coordinates: (42, )Example 2: Express the polar form (42, ) in rectangular coordinates:

Given: = 42 = = cos = 42 cos = 42 =4 = sin = 42 sin = 42 = 4So, the coordinates of the point are as follows:

Polar Coordinates: (42, ) Rectangular coordinates: (4, 4)

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Trigonometry

Polar Coordinates (contd)

Expressing Complex Numbers in Polar Form

A complex number can be represented as point in the Cartesian Plane, using the horizontal axis for

the real component of the number and the vertical axis for the imaginary component of the number.

If we express a complex number in rectangular coordinates as = + ,we can also express it inpolar coordinates as =(cos + sin), with [0,2). Then, the equivalences between thetwo forms for are:

Convert Rectangular to Polar Convert Polar to Rectangular

Magnitude: |

| =

=

+

x-coordinate:

=

cos

Angle: = tan y-coordinate: = sin Since will generally have two values on [0,2), you need to be careful to select the angle in thequadrant in which = + resides.Operations on Complex Numbers in Polar Form

Another expression that may be useful is: = cos + sin,a complex number can be expressedas an exponential form of . That is: = + =(cos + sin) = It is this expression that is responsible for the following rules regarding operations on complex

numbers. Let: = + =(cos + sin), = + =(cos + sin). Then,Multiplication: =[cos ( + ) + sin( + )]

So, to multiply complex numbers, you multiply their magnitudes and add their angles.

Division: = [cos ( ) + sin( )]So, to divide complex numbers, you divide their magnitudes and subtract their angles.

Powers: =(cos + sin)This results directly from the multiplication rule.

Roots: = cos + sin also, see DeMoivres Theorem belowThis results directly from the power rule if the exponent is a fraction.

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Trigonometry

DeMoivres Theorem

Abraham de Moivre (1667-1754) was a French mathematician who provided us with a very useful

Theorem for dealing with operations on complex numbers.

If we let =(cos + sin), DeMoivres Theorem gives us the power rule expressed on the priorpage: =(cos + sin )

Example 1: Find 3 + 76First, since = + , we have =3 and =7.Then, =(3) + (7) = 4; 6 = 46 = 4,096And, = tan = 138.590 in QII

6 = 831.542 ~ 111.542So,

3 + 76 = 4,096 [cos(111.542) + sin(111.542)]=1,504.0 + 3,809.9

Example 2: Find 5 25First, since = + , we have =2 and =7.Then, =(5) + (2) = 3; 5 = 35 = 243And, = tan 5 = 221.810 in Q III

5 = 1,109.052 ~ 29.052So,5 25 = 243 [cos(29.052) + sin(29.052)]= 212.4 + 118.0

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Trigonometry

DeMoivres Theorem for Roots

Let

=

(cos

+

sin

). Then,

has

distinct complex

-th roots that occupy positions

equidistant from each other on a circle of radius . Lets call the roots: , , , Then, theseroots can be calculated as follows: = cos + (2) + sin + (2)

The formula could also be restated with 2replaced by 360if this helps in the calculation.Example: Find the fifth roots of .

First, since = + , we have = 2 and =3.Then, =2 + (3) =13; =13 ~ 1.2924And, = tan =56.310; 5 =11.262The incremental angle for successive roots is: 360 5 roots = 72.

Then create a chart like this:

Fifth roots of () = ~ . =.

Angle(

)

=

+

0 11.262 = 1.2675 0.2524 1 11.262 + 72 = 60.738 = 0.6317 + 1.1275 2 60.738 + 72 = 132.738 = 0.8771 + 0.9492 3 132.738 + 72 = 204.738 = 1.1738 0.5408 4 204.738 + 72 = 276.738 = 0.1516 1.2835

Notice that if we add another 72, we get 348.738, which is equivalent to our first angle,

11.262because348.738

360 =

11.262. This is a good thing to check. The next

angle will always be equivalent to the first angle! If it isnt, go back and check your work.

Roots fit on a circle: Notice that, since all of the roots of have the same magnitude, and their angles that are 72apart from

each other, that they occupy equidistant positions on a circle with

center (0, 0) and radius =13 ~ 1.2924.

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Trigonometry

Graphing Polar Equations The Cardioid

Example: = + This cardioid is also a limaon of form = + sinwith =. The use of the sine functionindicates that the large loop will be symmetric about the-axis. The +sign indicates that the largeloop will be above the -axis. Lets create a table of values and graph the equation:

The four Cardioid forms:

= + 0 2

/6 3 7/6 1/3

3.732 4/3 0.268/2 4 3/2 0

2/3 3.732 5/3 0.2685/6 3 11/6 1 2 2 2

Once symmetry is

established, these values

are easily determined.

Generally, you want to look at

values of in [0,2]. However,some functions require larger

intervals. The size of the interval

depends largely on the nature of the

function and the coefficient of

.

Orange points on the

graph correspond to

orange values in the table.

Blue points on the graph

correspond to blue values

in the table.

The portion of the graph

above the x-axis results

from in Q1 and Q1,where the sine function ispositive.

Similarly, the portion of

the graph below the x-axis

results from in Q3 andQ4, where the sine

function is negative.

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Trigonometry

Graphing Polar Equations The Rose

Example: = This function is a rose. Consider the forms = sin and = cos .The number of petals on the rose depends on the value of .

If is an even integer, the rose will have 2petals. If is an odd integer, it will have petals.

Lets create a table of values and graph the equation:

The four Rose forms:

= 0 0/12 2 7/12 2/6 3.464 2/3 3.464/4 4 3/4 -4/3 3.464 5/6 3.464

5/12 2 11/12 2/2 0 0

Once symmetry is

established, these values

are easily determined.

Because this function involves an

argument of 2,we want to start bylooking at values of in [0, 2] 2 = [0,]. You could plot morepoints, but this interval is sufficientto establish the nature of the curve;

so you can graph the rest easily.

Orange points on the

graph correspond to

orange values in the table.

Blue points on the graph

correspond to blue values

in the table.

The values in the table

generate the points in the

two petals right of the -axis.Knowing that the curve is a

rose allows us to graph the

other two petals without

calculating more points.

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Trigonometry

Vectors

A vector is a quantity that has both magnitude and direction. An example would be wind blowing

toward the east at 30 miles per hour. Another example would be the force of 10 kg weight beingpulled toward the earth (a force you can feel if you are holding the weight).

Special Unit Vectors

We define unit vectors to be vectors of length 1. Unit vectors having the direction of the positive

axes will be quite useful to us. They are described in the chart and graphic below.

Unit Vector Direction

positive -axis positive -axis positive -axis

Vector Components

The length of a vector, , is called its magnitude and is represented by the symbol . If a vectorsinitial point (starting position) is (,), and its terminal point (ending position) is (,), then thevector displaces

= in the

-direction and displaces

= in the

-direction. We

can, then, represent the vector as follows:

= + The magnitude of the vector, , is calculated as:

= + If this looks familiar, it should. The magnitude of a vector is determined as the length of the

hypotenuse of a triangle with sides and using the Pythagorean Theorem.In three dimensions, tese concepts expand to the following:

= + + = + +

Similarly, vectors can be expanded to any number of dimensions.

Graphical

representation ofunit vectors andjin two dimensions.

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Trigonometry

Vector Properties

Vectors have a number of nice properties that make working with them both useful and relatively

simple. Let and be scalars, and let u, v and wbe vectors. Then, If = + , then = cos and = sin Then, = cos + sin (note: this formula is used in Force calculations) If =+ and = + , then + = ( + ) + ( + ) If = + , then = () + () Define to be the zero vector(i.e., it has zero length, so that = = 0). Note: the zero

vector is also called the null vector.

Note: = + can also be shown with the following notation: =, . This notation will beuseful in calculating dot products and performing operations with vectors.

Properties of Vectors

+ = + = Additive Identity

+ (

) = (

) +

=

Additive Inverse

+ = + Commutative Property + ( + ) = ( + ) + Associative Property () = () Associative Property ( + ) = + Distributive Property

( + ) = + Distributive Property 1(

) =

Multiplicative Identity

Also, note that:

= || Magnitude Property

Unit vector in the direction of

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General Example

, 4,3

,

2,

2

+ 8 + 6 = 14

alternative

vector

notation

Trigonometry

Vector Dot Product

The Dot Product of two vectors,

=

+

and

=

+

, is defined as follows:

= ( ) + ( )It is important to note that the dot product is a scalar, not a vector. It describes something about the

relationship between two vectors, but is not a vector itself. A useful approach to calculating the dot

product of two vectors is illustrated here:

= + =, = + =, Take a look at the example at right. Notice that thetwo vectors are lined up vertically. The numbers in

the each column are multiplied and the results are

added to get the dot product. So in this example, 4,3 2,2 = 14.Properties of the Dot Product

Let be a scalar, and let u, v and wbe vectors. Then,

=

= 0 Zero Property

= = 0 The same property holds in 3Dfor any pair of , , and = Commutative Property = Magnitude Square Property ( + ) = ( ) + ( ) Distributive Property ( ) = () = () Multiplication by a Scalar Property

More properties:

If = 0and and , thenandare orthogonal (perpendicular). If there is a scalar such that =, then andare parallel. If is the angle between and, then cos =

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Trigonometry

Vector Dot Product (contd)

Vector Projection

The projection of a vector, , onto another vector , is obtained using the dot product. The formulaused to determine the projection vector is:

proj = Notice that

is a scalar, and that projis a vector.In the diagram at right, v1= proj

.

Orthogonal Components of a Vector

A vector, , can be expressed as the sum of two orthogonal vectors and, as shown in the abovediagram. The resulting vectors are:

= proj = and =

Work

Work is a scalar quantity in physics that measures the force exerted on an object over a particular

distance. It is defined using vectors, as shown below. Let:

Fbe the force vector acting on an object, moving it from pointto point . be the vector fromto .

be the angle between Fand

.

Then, we define work as:

= = cos

v

v1 w

v2

Both of these formulas are useful.

Which one you use in a particular

situation depends on what

information is available.

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Page Subject

30 AmbiguousCase

for

Oblique

Triangles

32 AmbiguousCaseforObliqueTriangles Flowchart

26 AngleAdditionFormulas

AreaofaTriangle

33 GeometryFormula

33 Heron'sFormula

34 TrigonometricFormulas

34 CoordinateGeometryFormula

39 Cardioid

28 Cofunctions

36 ComplexNumbers OperationsinPolarForm

36 ComplexNumbers

in

Polar

Form

41 ComponentsofVectors

41 ConversionbetweenRectangularandPolarCoordinates

7 CosecantFunction

7 CosineFunction

7 CotangentFunction

7 DefinitionsofTrigFunctions(RightTriangle)

6 DefinitionsofTrigFunctions(x andy axes)

37 DeMoivre'sTheorem

38 DeMoivre'sTheoremforRoots

43 DotProduct

26 DoubleAngleFormulas

Graphs

10 BasicTrigFunctions

39 Cardioid

22 CosecantFunction

14 CosineFunction

18 CotangentFunction

25 InverseTrigonometricFunctions

40 Rose

20 SecantFunction

12 Sine

Function16 TangentFunction

11 TableofTrigFunctionCharacteristics

26 Half AngleFormulas

33 Heron'sFormula


Index

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Page Subject


Index

InverseTrigonometric

Functions

24 Definitions

25 Graphs

24 PrincipalValues

24 Ranges

28 LawofCosines

28 LawofSines

29 ObliqueTriangle MethodstoSolve

36 OperationsonComplexNumbersinPolarForm

44 OrthogonalComponentsofaVector

35 PolarCoordinates

36 Polarform

of

Complex

Numbers

in

Polar

Form

35 PolartoRectangularCoordinateConversion

27 PowerReducingFormulas

24 PrincipalValuesofInverseTrigonometricFunctions

27 ProducttoSumFormulas

44 ProjectionofOneVectorontoAnother

42 PropertiesofVectors

28 PythagoreanIdentities

6 Radians

42 RectangulartoPolarCoordinateConversion

40 Rose

7 SecantFunction

7 SineFunction

6 SineCosineRelationship

7 SOHCAHTOA

27 SumtoProductFormulas

7 TangentFunction

8 TrigFunctionValuesinQuadrantsII,III,andIV

7 TrigFunctionsofSpecialAngles

9 UnitCircle

41 UnitVectors iandj

41 Vectors44 SpecialUnitVectors iandj

41 VectorComponents

42 VectorProperties

43 DotProduct

44 VectorProjection

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trigonometry handbook (formula)

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