Trigonometry Essentials Practice Workbook with Answers: Master Basic Trig Skills

191 Pages·1.826 MB·English

by  Chris McMullen

Trigonometry Essentials
Practice Workbook
with Answers
Master Basic Trig Skills
Improve Your Math Fluency Series
ChrisMcMullen,Ph.D.
TrigonometryEssentialsPracticeWorkbookwithAnswers: MasterBasicTrigSkills
ImproveYourMathFluencySeries
Copyright ©2012,2015,2017ChrisMcMullen,Ph.D.
All rights reserved. This includes the right to reproduce any portion of this book in any
form. However,teacherswhopurchaseonecopyofthisbook,orborrowonephysicalcopy
from a library, may make and distribute photocopies of selected pages for instructional
purposes for their own classes only. Also, parents who purchase one copy of this book, or
borrow one physical copy from a library, may make and distribute photocopiesof selected
pagesforusebytheirownchildrenonly.
CreateSpace
Professional&Technical/Science/Mathematics/Trigonometry
Professional&Technical/Education/SpecificSkills/Mathematics/Trigonometry
PrinteditionISBN: 1477497781
PrinteditionEAN-13: 978-1477497784
TrigonometryEssentialsPracticeWorkbookwithAnswers
Contents
MakingtheMostofthisWorkbook 4
Chapter1: ConvertingDegreestoRadians 5
Chapter1Answers 19
Chapter2: ConvertingRadianstoDegrees 21
Chapter2Answers 34
Chapter3: IdentifyingTrigFunctionsinRightTriangles 36
Chapter3Answers 51
Chapter4: SpecialRightTriangles 52
Chapter4Answers 68
Chapter5: MemorizeBasicTrigFunctionsinQuadrantI 69
Chapter5Answers 85
Chapter6: FindingtheReferenceAngle 87
Chapter6Answers 102
Chapter7: DetermineBasicTrigFunctionsinQuadrantsII-IV 104
Chapter7Answers 119
Chapter8: TheInverseTrigFunctions 121
Chapter8Answers 135
Chapter9: TheLawofSinesandtheLawofCosines 137
Chapter9Answers 156
Chapter10: LearnandApplyTrigIdentities 157
Chapter10Answers 180
Chapter11: SolveAlgebraicEquationsthatInvolveTrigFunctions 181
Chapter11Answers 191
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Making the Most of this Workbook
 Mathematics is a language. You can’t hold a decent conversation in any language if
youhavealimitedvocabularyorifyouarenotfluent. Inordertobecomesuccessful
inmathematics,youneed topracticeuntilyouhavemasteredthefundamentalsand
developed fluency in the subject. This Trigonometry Essentials Practice Workbook
with Answers: Master Basic Trig Skills will helpyou improve the fluency with which
you apply fundamental trig techniques. Every problem can be answered without a
calculator, which isvery helpful for studentswho aren’t allowed to use a calculator.
This is the case in some trig and physics courses, as well as some standardized
exams(liketheMCAT).
 Thisworkbookisconvenientlydividedinto11chapterssothatyoucanfocusonone
basic skill at a time. The first two chapters provide practice converting between
degrees and radians. Chapters 3-4 are devoted toward relating the basic trig
functions to right triangles; Chapter 4 focuses on the - - and - -
triangles. StudentsmasterthebasictrigfunctionsinChapters5-7,andtheirinverse
functions in Chapter 8. Chapter 9 is devoted to the law30o°f6si0n°e9s0a°nd law45o°f c4o5s°in9e0s°,
and includes several obtuse and acute triangles for practice. Trig identities and
their application are the subject of Chapter 10, and Chapter 11 focuses on how to
solveequationsthatfeaturetrigfunctions.
 Eachchapterbeginswithconciseinstructionsdescribinghowtoperformabasictrig
skill – such as how to determine the reference angle. These instructions are
followed by a few examples. Use these examplesasa guide untilyoubecome fluent
inthetechnique.
 After you complete a page, check your answers with the answer key in the back of
the book. Practice makes permanent, but not necessarily perfect: If you practice
making mistakes, you willlearn yourmistakes. Check your answersand learn from
yourmistakes such that youpractice solvingthe problemscorrectly. Thisway your
practicewillmakeperfect.
 Math can be fun. Make a game of your practice by recording your times and trying
to improve on your times, and recordingyour scoresand trying toimprove on your
scores. Doing this will help you see how much you are improving, and this sign of
improvement can give you the confidence to succeed in math, which can help you
learntoenjoythissubjectmore.
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TrigonometryEssentialsPracticeWorkbookwithAnswers
Chapter 1: Converting Degrees to Radians
There are two commonmethodsfor measuringangles. One method isto divide a circle up
into slices, where each slice equals one degree ( ). In the degree measure, a full circle
corresponds to , a right angle is , an equilateral triangle has angles, and so on.
Degr3e6es0are very common in science and engineerin°g, since a protractor is typically ruled
indegrees. 360° 90° 60°
1°
60°
90°
60° 60°
360°
A second method for measuring angles is to work with radians instead of degrees.
Radians are defined such that radians correspond to a full circle. The unit radian is
often abbreviated rad. In terms of radians, a right angle is rad, an equilateral triangle
has angles of rad, and so o2nπ. Radians are very common in math courses since many
geometric formulas involve π. π/2
π/3
rad
1 rad
π/3
rad rad rad
π/3 π/3
π/2
rad
2π
Since both unitsfor angular measure – degrees and radians – are very common, it’s
usefultobeabletoconvertdegreestoradiansorvice-versa.
Themainideabehindtheconversionisgeometric. Forexample, isone-sixthofa
circle, since equals . Therefore, the same angle in radiansis rad, since one-
sixthof is ,whichreducesto . Similarly, isone-fourthof6a0°circle,since
dividedby 6e0q°uals .36T0h°is/6equatesto rad,since dividedby reπd/u3cesto .
2π 2π/6 π/3 90° 360°
4 90° π/2 2π 4 π/2
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Theconversionfactorneededtoconvertdegreestoradiansorvice-versais:
radians
The reason for this is that a full circl1e8i0s° = πor radians. Therefore, is equivalent
to radians. Ifwedividebothquantitiesby ,wefindthat equatesto radians.
Dividing both sides by , you can3s6e0e°tha2tπ radian equates to ap3p6r0o°ximately
deg2rπees. However,it’susuallymoreconvenien2ttoremember18th0a°t rπadiansthanto
memorizethenumber . π 1 57.3
Inthischapter,we willpracticeconvertingdegreesintorad1ia8n0s°.=Thπewaytodothis
istomultiplyby radia5n7s.3anddivideby ,asillustratedinthefollowingexamples.
Take some time to understand these concepts, and study the examples. Once you
understand the fπollowing examples, you1a8r0e°ready to practice the technique yourself. You
may need to refer to the examples frequently as you begin, but should try to solve the
exercises all by yourself once you get the hang of it. Be sure to check the answers at the
backofthebooktoensurethatyouaresolvingtheproblemscorrectly.
Instructions: Convertthegivenanglefromdegreestoradians.
Procedure: Multiply thegiven angle by and divide by . If the result isa fraction, see
if the fraction is reducible. That is, if the numerator and denominator are both evenly
divisible by an integer greater than 1, thπen the fraction i1s8r0educible. Reduce a fraction by
dividing both the numerator and denominator by the greatest common factor (as in the
examplesthatfollow).
Example1:
rad
rad rad
120°
Both the numerator ( ) and dπen omina1t2o0r ( ) a2reπ divisible by : ,
120°× = π =
. Theansweris ra1d8. 0° 180 3
120 180 60 120/60= 2
1E8xa0m/6p0le=23: 2π/3
rad
rad rad
30°
Both the numerator ( ) and deπn ominat3o0r ( ) aπre divisible by : ,
30°× = π =
. Theansweris rad1. 80° 180 6
30 180 30 30/30 = 1
1E8xa0m/3p0le=36: π/6
rad
rad rad
720°
Boththenumerator( )anddenoπm inator7(20 )are divisibleby : . The
720°× = π = 4π
answeris rad. 180° 180
720 180 180 720/180 = 4
4π
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TrigonometryEssentialsPracticeWorkbookwithAnswers
Instructions: Convertthegivenanglefromdegreestoradians. Checkyouranswersinthe
backofthebook.
(1)36° (2)18°
(3)780° (4)117°
(5)45° (6)12°
(7)96° (8)270°
(9)120° (10)78°
(11)48° (12)150°
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Instructions: Convertthegivenanglefromdegreestoradians. Checkyouranswersinthe
backofthebook.
(1)210° (2)45°
(3)24° (4)3°
(5)10° (6)135°
(7)96° (8)315°
(9)270° (10)42°
(11)87° (12)69°
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TrigonometryEssentialsPracticeWorkbookwithAnswers
Instructions: Convertthegivenanglefromdegreestoradians. Checkyouranswersinthe
backofthebook.
(1)153° (2)48°
(3)57° (4)63°
(5)96° (6)552°
(7)606° (8)132°
(9)225° (10)300°
(11)129° (12)60°
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Instructions: Convertthegivenanglefromdegreestoradians. Checkyouranswersinthe
backofthebook.
(1)348° (2)117°
(3)105° (4)816°
(5)90° (6)57°
(7)27° (8)69°
(9)525° (10)276°
(11)63° (12)636°
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