1

NAME: ___________________ PERIOD: ___ DATE: ______________ MATH ANALYSIS 2

MR. MELLINA

CHAPTER 10: ADDITIONAL TRIG FORMULAS Sections:

v 10.1 Formulas for 𝑐𝑜𝑠 𝛼 ± 𝛽 and 𝑠𝑖𝑛 𝛼 ± 𝛽 v 10.2 Formulas for 𝑡𝑎𝑛 𝛼 ± 𝛽

v 10.3 Double Angle and Half Angle Formulas v 10.4 Solving Trigonometric Equations

HWSets

SetA(Section10.1)Page373&374,#’s2-10even,14,18,20,24,28,34-38even.SetB(Section10.2)Page377,#’s2-8even,12,14,18,20,24.SetC(Section10.3)Pages383&384,#’s2-10even,14-24even,32-44even.SetD(Section10.4)Page390,#’s10-32even.

2

10.1 FORMULAS FOR 𝐜𝐨𝐬 𝜶 ± 𝜷 AND 𝐬𝐢𝐧 𝜶 ± 𝜷 (PAGE 369) Objectives:Toderiveandapplytheformulasfor𝑐𝑜𝑠 𝛼 ± 𝛽 andfor𝑠𝑖𝑛 𝛼 ± 𝛽 . 10.1WarmUp!Expresstheangleasasumordifferenceofspecialangles.a. 15° b. 75° c. 165° d. 2

34 e. 52

34 f. 62

34

Evaluatethefollowingwithoutacalculator.g. sin 45° h. sin 75°Example1Findanexampletoshowthat,ingeneral,eachstatementistrue.a. sin 𝑥 + 𝑦 ≠ sin 𝑥 + sin 𝑦 b. sin 𝑥 + 𝑦 = sin 𝑥 + sin 𝑦

Sum and Difference Formulas for Cosine and Sine

sin(𝛼 + 𝛽) = sin 𝛼 cos 𝛽___ cos 𝛼 sin 𝛽 sin(𝛼 − 𝛽) = sin 𝛼 cos 𝛽___ cos 𝛼 sin 𝛽 cos(𝛼 + 𝛽) = cos 𝛼 cos 𝛽___ sin 𝛼 sin 𝛽 cos(𝛼 − 𝛽) = cos 𝛼 cos 𝛽___ sin 𝛼 sin 𝛽

2 main purposes for using the Addition and Subtraction Formulas

1. Finding exact values of trigonometric expressions. 2. Simplifying expressions to obtain other identities.

3

Example2Simplifyeachexpression.a. sin 1° cos 2° + cos 1° sin 2° b. cos 2

Lcos 2

L− sin 2

Lsin 2

L

c. sin 20° cos 15° − cos 20° sin 15° d. cos 75° cos 25° + sin 75° sin 25° e. sin 75° cos 15° + cos 75° sin 15° f. cos 52

34cos 2

34− sin 52

34sin 2

34

g. sin 3𝑥 cos 2𝑥 − cos 3𝑥 sin 2𝑥Example3Provethatthegivenequationisanidentitya. sin 𝑥 + 𝜋 = − sin 𝑥 b. cos 𝑥 + 2

4= − sin 𝑥

4

Example4Findtheexactvalueofeachexpression.a. cos75° b. sin(-15°) c. cos105° d. sin 62

34

Example5Findtheindicatedsumordifferencea. Supposethatsin 𝛼 = P

5andsin 𝛽 = 4L

45where0 < 𝛼 < 2

4< 𝛽 < 𝜋.Findsin 𝛼 + 𝛽 .

Example6Simplifytheexpression a. cos 𝑥 cos 𝑦 tan 𝑥 + tan 𝑦

5

Example7IndependentWork**Findtheindicatedsumordifference

a. Supposethattan 𝛼 = LPandtan 𝛽 = 34

5,where0 < 𝛼 < 𝛽 < 2

4.Findcos 𝛼 − 𝛽

Example8IndependentWork***Simplifythegivenexpression.a. sin 30° + 𝜃 + sin 30° − 𝜃 b. cos 2

P+ 𝑥 + cos 2

P− 𝑥

c. cos P2

4+ 𝑥 − cos P2

4− 𝑥

6

d. VWX YZ[ ZVWX Y\[VWXY VWX[

e. sin 𝑥 + 𝑦 sec 𝑥 sec 𝑦

7

10.2 FORMULAS FOR 𝐭𝐚𝐧 𝜶 ± 𝜷 (PAGE 375) Objectives:Toderiveandapplyformulasfor𝑡𝑎𝑛 𝛼 ± 𝛽 10.2WarmUp!Tellwhetherthetwogivenlinesareparallel,perpendicular,orneither.a. 2𝑥 + 5𝑦 = 5 b. 3𝑥 + 2𝑦 = 5 5𝑥 − 2𝑦 = 10 3𝑦 + 2𝑥 = −3 Evaluatec. tan 45° d. tan 75°Example1Findtan 𝛼 + 𝛽 andtan 𝛼 − 𝛽 .a. tan 𝛼 = 4

Pandtan 𝛽 = 3

4

Example2Findtheexactvalueofthegivenexpression.

a. `aX 65°\`aX Pb°3Z`aX 65° `aX Pb°

b.`aXcde Z`aX

dfc

3\`aXcde `aXdfc

Sum and Difference Formulas for Tangent

Sum: tan(𝛼 + 𝛽) = `aXY `aX[3 `aXY `aX [

Difference: tan(𝛼 − 𝛽) = `aX Y`aX [3 `aX Y `aX[

8

Example3Usethegiveninformationtoevaluate.a. Evaluatetan 2

L+ 𝜃 whentan 𝜃 = 3

4

b. Evaluatetan 75° c. Evaluatetan 165° Example4Findthetwosupplementaryanglesformedbythetwolinesgiven.Roundtothenearesttenth.a. 𝑦 − 3𝑥 = −5

−𝑥 + 𝑦 = 4

Angles Between 2 Lines tan 𝜃 =

−1 +

where 𝑚3 and 𝑚4 are the slopes of the two given lines.

9

Example5Usethegiveninformationtofindthetangentsumordifference.a. Supposethattan 𝛼 = 3

Landtan 𝛽 = P

5where0 < 𝛼 < 𝛽 < 2

4.Findtan 𝛼 + 𝛽 .

b. Supposethat𝛼 = tan\3 2 and𝛽 = tan\3 3 .Showthattan 𝛼 + 𝛽 = −1.c. Supposecot 𝛼 = 2andcot 𝛽 = 4

P,where0 < 𝛼 < 𝛽 < 2

4Findcot 𝛼 + 𝛽 .

10

d. Supposesin 𝛼 = P5andcos 𝛽 = 5

3Pwhere0 < 𝛼 < 𝛽 < 2

4Find:

sin 𝛼 + 𝛽 cos 𝛼 + 𝛽 tan 𝛼 + 𝛽

Example6:IndependentWork**GivenA(3,1),B(14,-1),andC(5,5),findthemeasureof< 𝐵𝐴𝐶by:a. TheslopesoflinesABandAC b. Thelawofcosines

11

10.3 DOUBLE-ANGLE AND HALF-ANGLE FORMULAS (PAGE 380) Objectives:Toderiveandapplydouble-angleandhalf-angleformulas. 10.3WarmUp!Substitute𝛽with𝛼inthefollowingequationsandsimplify.a. sin 𝛼 + 𝛽 = sin 𝛼 cos 𝛽 + cos 𝛼 sin 𝛽 b. cos 𝛼 + 𝛽 = cos 𝛼 cos 𝛽 − sin 𝛼 sin 𝛽 c. tan 𝛼 + 𝛽 = `aXYZ`aX[

3\`aXY `aX[ d. Evaluate:cos(67.5°)

Example1Simplifythegivenexpressiona. 2 sin 10° cos 10° b. cos4 15° − sin4 15° c. 1 − 2 sin4 35°d. 2 cos4 25° − 1 e. 4 `aX 5b°

3\`aXc 5b° f. 1 − sin4 𝑥

Double-Angle Formulas Half-Angle Formulas

sin 2𝛼 = sin 𝛼 cos𝛼 sin Y4= ±l3 mnVY

4

cos 2𝛼 = cos4 𝛼 sin4 𝛼 cos Y4= ±l3 mnVY

4

= 1 2sin4 𝛼

= 2 cos4 𝛼 1 tan Y4= ±l3 mnVY

3 mnVY= VWX Y

3 mnVY= 3 mnVY

VWX Y

tan 2𝛼 =tan 𝛼1 tan4 𝛼

12

g. 2 sin 3𝛼 cos 3𝛼 h. 2 cos4 10° − 1 i. L `aX[3\`aXc [

j. 2 sin 35° cos 35° k. 1 − 2 sin4 o4 l. 3ZmnV 6b°

4

Example2Findtheexactvalueofthegivenexpressiona. 2 cos4 2

p− 1 b. cos4 2

34− sin4 2

34

c. sin 15° cos 15°Example3InthefollowingproblemsangleAisacute.a. Ifsin 𝐴 = 5

3P ,findsin 2𝐴andcos 2𝐴. b. Ifsin 𝐴 = P

5findsin 2𝐴andsin 4𝐴.

13

c. Ifcos 𝐴 = 35findcos 2𝐴andcos q

4.

Example4Findthefollowingusingadouble-angleorhalf-angleformula.a. cos105° b. sin 117.5°c. tan 15° f. cos 67.5°

14

Example5Provethatthegivenequationisanidentity

a. VWX 4q3\mnV 4q

= cot 𝐴 b. sin o4+ cos o

4

4= 1 + sin 𝑥

c. 3\`aXc o

3Z`aXc o= cos 2𝑥

Example6:IndependentWork**Simplifythegivenexpressiona. 3ZmnV 4o

mn` o b. 1 − sin4 𝑥 1 − tan4 𝑥

15

c. VWX PoVWX o

− mnV PomnV o

Example7:IndependentWork**Findtheexactvalueusingahalf-angleordouble-angleformula.a. tan 2

p b. cos 105°

d. sin 195° e. sin 22.5° g. cos − 52

34 h. cos − 62

34

16

10.4 SOLVING TRIGONOMETRIC EQUATIONS (PAGE 386) Objectives:Touseidentitiestosolvetrigonometricequations. 10.4WarmUp!Solvethefollowingequationswithsolutionsintheinterval0 ≤ 𝑥 < 2𝜋.a. sin 𝑥 = 0 b. 2 sin 𝑥 − 1 = 0 Example1Solvetheequationforsolution(s)inthegiveninterval.a. cos 2𝑥 = 1 − sin 𝑥,for0 ≤ 𝑥 < 2𝜋

Methods for Solving the Trigonometric Equation f(x) = g(x) 1. _______________: Use a graphing calculator or computer to graph both functions on the same set of axes. Use the intersection feature to find the x-coordinates of any intersection points of the two graphs within the given interval. 2. _______________: Use the following guidelines

• It may be helpful to draw a quick sketch of both functions to see roughly where the solutions are. • If the equation involves functions of 2x and x, transform the functions of 2x into functions of x by

using identities. • If the equation involves functions of 2x only, it is usually better to solve for 2x directly and then

solve for x. • Be careful not to lose roots when you divide both sides of an equation by a function of the variable.

17

b. 3 cos 2𝑥 + cos 𝑥 = 2,for0 ≤ 𝑥 < 2𝜋 c. 2 sin 2𝑥 = 1,for0 ≤ 𝑥 < 360°d. Useacalculatortofindthesmallestpositiverootofsin 2.8𝑥 = cos 𝑥for0 ≤ 𝑥 < 2𝜋. Example2Chooseamethodyouwouldchoosetosolvealgebraically.a. cos 2𝑥 = cos 𝑥 b. sin4 𝑥 = sin 𝑥 c. sin 𝑥 = cos 𝑥 d. sin 2𝑥 = cos 2𝑥 e. sin 3𝑥 = cos 3𝑥 f. tan 𝑥 − 10° = 1g. sin 4𝑥 = sin 2𝑥 h. cos 4𝑥 = 1 − 3 cos 2𝑥

18

Example3Solvetheequationforthesolution(s)in0 ≤ 𝑥 < 360°.a. cos 2𝑥 = cos 𝑥 b. sin 𝑥 = cos 𝑥 c. sin 3𝑥 = cos 3𝑥 d. sin 4𝑥 = sin 2𝑥

19

Example4Solveeachinequalityfor0 ≤ 𝑥 < 2𝜋byusingagraphingcalculator.Giveanswerstothenearesthundredth.a. cos 𝑥 > 3

4sin 𝑥 b. sin 3𝑥 − 2

4> 0 c. cos 𝑥 ≤ sin 2𝑥

Example5:IndependentWorkSolveeachequationfor0 ≤ 𝑥 < 360°byusingtrigonometricidentities.Giveanswerstothenearesttenthwhennecessary.a. 2 cos 𝑥 + 45° = 1 b. sin 60° − 𝑥 = 2 sin 𝑥c. sin 𝑥 = sin 2𝑥

20

Example6:IndependentWorkSolveeachequationfor0 ≤ 𝑥 < 2𝜋byusingtrigonometricidentities.Giveanswerstothenearesthundredthwhennecessary.a. sin 𝑥 cos 𝑥 = 3

4 b. tan 2𝑥 = 3 tan 𝑥

c. cos 2𝑥 = 5 sin4 𝑥 − cos4 𝑥 d. 3 sin 𝑥 = 1 + cos 2𝑥 e. cos 2𝑥 = sec 𝑥

21

10.1&10.2ReviewFornumbers1-7:Findtheexactvalue.1. sin 2

34 2. cos(15°)

3. cos(225°) 4. sin(75°)5. `aX 36b\`aX 5b°

3Z`aX 36b° `aX 5b° 6. sin 52

34cos 2

w− cos 52

34sin 2

w

7. tan 345°

22

Fornumber8-16:Simplify 8. tan 𝜃 + 3𝜋 9. sin 𝑥 cos 𝑥 + 𝑦 − cos 𝑥 sin 𝑥 + 𝑦 10. cos 2

P+ 𝑥 + cos 2

P− 𝑥 11. cos 75° cos 15° − sin 75° sin 15°

12. sin 75° cos 15° − cos 75° sin 15° 13. sin 30° + 𝑥 + sin 30° − 𝑥 14. cos 45° − 𝑥 − cos 45° + 𝑥 15. cos 250° cos 40° + sin 250° sin 40°16. sin 2

wcos 52

w+ sin 52

wcos 2

w

23

Fornumbers17&18:Findthetwosupplementaryanglesbetweenthelinesprovided17. 𝑦 = − 3

4𝑥 + 1 18. 𝑦 = 2𝑥 + 3

𝑦 = LP− 3𝑥 𝑦 = 1 − 3𝑥

Fornumbers19-21:Showthatthefollowingistrue.19. cos 𝜋 − 𝑥 = −cos 𝑥 20. cos 2

4+ 𝜃 = − sin 𝜃

21. Iftan 𝛼 = 3

Pandtan 𝛽 = 3

4,showthattan 𝛼 + 𝛽 = 1

Fornumbers22&23:Find𝐭𝐚𝐧 𝜶 + 𝜷 and𝐭𝐚𝐧 𝜶 − 𝜷 .22. tan 𝛼 = 2 23. tan 𝛼 = 2 tan 𝛽 = − 3

L tan 𝛽 = 3

L

24

24. Findtan 62L− 𝜃 whentan 𝜃 = 3

P 25. Findtan P2

L− 𝑥 whentan 𝑥 = 3

4

Fornumbers26&27:Supposethatsin 𝛼 = 𝟐

𝟐andcos 𝛽 = 𝟏𝟓

𝟏𝟕,where2

4< 𝛼 < 𝜋and

P24< 𝛽 < 2𝜋.Find:

26. sin 𝛼 − 𝛽 27. cos 𝛼 + 𝛽 .Fornumbers28&29:Supposethatsin 𝐴 = 𝟓

𝟏𝟑andcos 𝐵 = 𝟏𝟓

𝟏𝟕,where0 < 𝐵 < 2

4< 𝐴 <

𝜋.Find:28. sin 𝐴 + 𝐵 29. tan 𝐴 − 𝐵

25

Chapter10TestReview5. Explainwhytheformulafortan 𝛼 + 𝛽 cannotbeusedtosimplifytheexpression

tan 24+ 𝜃 .Howelsecouldthisexpressionbesimplified?Tryyourideaandgivea

result.6. Suppose< 𝐴isacuteandcos 𝐴 = 5

3P.Findeachofthefollowing.

a. sinA b. cos2A c. sin2A d. sin4A7. Simplifythegivenexpression a. 3ZmnV 4o

VWX 4o b. 1 + cot4 𝑦 cos 2𝑦 + 1

c. `aX }

V~m }\3 d. cos4 o

L− sin4 o

L

26

8. Evaluatethegivenexpression. a. sin 2

34cos 2

34 b. 1 − 2 sin4 52

34

9. Provethatthegivenequationisanidentity. a. 1 + tan4 𝑥 1 + cos 2𝑥 = 2

b. mnV � mVm �`aX �Zmn` �

= sin4 𝜃 + cos 2𝜃11. Solvecos 2𝑥 = sin 𝑥 − 2for0 ≤ 𝑥 < 2𝜋.

27