Download - Trigonometry Handbook (Formula)
-
8/11/2019 Trigonometry Handbook (Formula)
1/46
Copyright20122014,EarlWhitney,RenoNV. AllRightsReserved
MathHandbook
ofFormulas,ProcessesandTricks
Trigonometry
Preparedby: EarlL.Whitney,FSA,MAAA
Version
1.05
September
12,
2014
-
8/11/2019 Trigonometry Handbook (Formula)
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.
-2-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
3/46
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
TrigonometryHandbook
TableofContents
-3-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
4/46
TrigonometryHandbook
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
-4-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
5/46
TrigonometryHandbook
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.
-5-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
6/46
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
-6-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
7/46
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 =
-7-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
8/46
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: =
-8-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
9/46
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.
-9-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
10/46
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
-10-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
11/46
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).
-11-
Version 1.05
-
8/11/2019 Trigonometry Handbook (Formula)
12/46
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: = =
-12-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
13/46
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:
+
,= (,)
Step 5: The -value of thepoint halfway between the
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.
-13-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
14/46
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
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: = =
-14-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
15/46
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:
+ ,=,
Step 4: The -value of thepoint halfway between the
left and center extrema is
"".
The point is:
+ ,= (,)
Step 5: The -value of thepoint halfway between the
center and right extrema 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 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.
Note: If 0, all pointson the curve are shifted
vertically by units.
-15-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
16/46
For this example:
Amplitude: = || = || = Period: = = =Phase Shift: = =
=
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
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 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).
-16-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
17/46
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:
= =
Step 4: The -value of thepoint halfway between the
zero point and the right
asymptote is "".
The point is:
+
,=,
Step 5: The -value of thepoint halfway between the
left asymptote and the zero
point is " ".
The point is:
+ ,=,
Step 7: Duplicate the wave
to the left and right as
desired.
Step 6: Draw a smooth
curve through the three keypoints, approaching the
asymptotes on each side.
This will produce the graphof one wave of the function.
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.
Note: If 0, all pointson the curve are shifted
vertically by units.
-17-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
18/46
For this example:
Amplitude: = || = || = Period: = = =Phase Shift: = =
=
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
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 cycle, shifted to the right, is shown in orange below.
Note that, for the
cotangent curve,
we typically graph
the asymptotes
first, and then
graph the curve
between them (see
next page).
-18-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
19/46
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: + ,= (,)
Step 4: The -value of thepoint halfway between the
left asymptote and the zero
point is "".
The point is:
+
,=,
Step 5: The -value of thepoint halfway between the
zero point and the right
asymptote is " ".
The point is:
+ ,=,
Step 7: Duplicate the wave
to the left and right as
desired.
Step 6: Draw a smooth
curve through the three key
points, approaching the
asymptotes on each side.
This will produce the graphof one wave of the function.
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.
Note: If 0, all pointson the curve are shifted
vertically by units.
-19-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
20/46
For this example:
Amplitude: = || = || = Period: = = =Phase Shift: = =
=
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
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 cycle, shifted to the right, is shown in orange below.
One cycle of the secant curve contains two U-shaped curves, one
opening up and one opening down.
-20-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
21/46
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.
Step 6: Duplicate the wave
to the left and right as
desired. Erase the cosine
function if necessary.
This will produce the graph
of one wave of the function.
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.
Note: If 0, all pointson the curve are shifted
vertically by units.
-21-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
22/46
For this example:
Amplitude: = || = || = Period: = = =Phase Shift: = =
=
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
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 cycle, shifted to the right, is shown in orange below.
One cycle of the cosecant curve contains two U-shaped curves, one
opening up and one opening down.
-22-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
23/46
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.
Step 6: Duplicate the wave
to the left and right as
desired. Erase the sine
function if necessary.
This will produce the graph
of one wave of the function.
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.
Note: If 0, all pointson the curve are shifted
vertically by units.
-23-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
24/46
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
-24-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
25/46
Trigonometry
Graphs of Inverse Trigonometric Functions
-25-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
26/46
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
-26-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
27/46
Trigonometry
Key Angle Formulas (contd)
Power Reducing Formulas
sin = cos cos = +cos tan = cos+cos
Product-to-Sum Formulas
= [( ) ( + )] = [( ) + ( + )] = [( + ) + ( )]
=
[
(
+
)
(
)]
Sum-to-Product Formulas
+ = + =
+
+ = + = +
-27-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
28/46
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
-28-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
29/46
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.
-29-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
30/46
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
-
8/11/2019 Trigonometry Handbook (Formula)
31/46
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
-
8/11/2019 Trigonometry Handbook (Formula)
32/46
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
-32-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
33/46
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.
-33-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
34/46
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:
= =
+ = =
-34-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
35/46
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)
-35-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
36/46
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.
-36-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
37/46
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
-37-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
38/46
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.
-38-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
39/46
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.
-39-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
40/46
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.
-40-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
41/46
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.
-41-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
42/46
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
-42-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
43/46
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 =
-43-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
44/46
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.
-44-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
45/46
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
TrigonometryHandbook
Index
-45-
Version 1.05 9/12/2014
-
8/11/2019 Trigonometry Handbook (Formula)
46/46
Page Subject
TrigonometryHandbook
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
-46-