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11. a) Determine graphically, using17. Determine an expression for m that- cos2 x m sin xmakes 2sin xan identity.technology, whether the expression2x - cot2 x appears to be equivalentcsccos xto csc x or sec x.b) What are the non-permissible values, inradians, for the identity from part a)?csc2 x - cot2 x as the singlec) Expresscos xreciprocal trigonometric ratio that youidentified in part a).π12. a) Substitute x into the equation4cot x sin x csc x to determinesec xwhether it could be an identity. Useexact values.b) Algebraically confirm that theexpression on the left side simplifiesto csc x.13. Stan, Lina, and Giselle are workingtogether to try to determine whether theequation sin x cos x tan x 1 is anidentity.a) Stan substitutes x 0 into each side ofthe equation. What is the result?πb) Lina substitutes x into each side of2the equation. What does she observe?c) Stan points out that Lina’s choice is notpermissible for this equation. Explainwhy.πd) Giselle substitutes x into each side4of the equation. What does she find?e) Do the three students have enoughinformation to conclude whether ornot the given equation is an identity?Explain.14. Simplify (sin x cos x)2 (sin x - cos x)2.Extend15. Given csc2 x sin2 x 7.89, find the value1 1 .ofcsc2 xsin2 x16. Show algebraically that11 2 sec2 θ1 sin θ1 - sin θis an identity.298 MHR Chapter 6Create ConnectionsC1 Explain how a student who does notknow the cot2 x 1 csc2 x form of thePythagorean identity could simplify anexpression that contained the expressionsin2 x .cot2 x 1 using the fact that 1 sin2 xC2 For some trigonometric expressions,multiplying by a conjugate helps tosimplify the expression. Simplifysin θby multiplying the numerator1 cos θand the denominator by the conjugateof the denominator, 1 - cos θ. Describehow this process helps to simplify theexpression.C3MINI LAB Explore theMaterialseffect of different domainson apparent identities. graphingcalculatorStep 1 Graph the two functionssin xy tan x and y cos x on thesame grid, using a domain ofπ . Is there graphical evidence0 x 2sin xthat tan x cos x is an identity? Explain.Step 2 Graph the two functions y tan x andsin xy cos x again, using the expandeddomain -2π x 2π. Is the equationsin xtan x cos x an identity? Explain.Step 3 Find and record a differenttrigonometric equation that is trueover a restricted domain but is not anidentity when all permissible valuesare checked. Compare your answerwith that of a classmate. Step 4 How does this activity show theweakness of using graphical andnumerical methods for verifyingpotential identities?

6.2Sum, Difference, andDouble-Angle IdentitiesFocus on applying sum, difference, and double-angle identities toverify the equivalence of trigonometric expressions verifying a trigonometric identity numerically and graphicallyusing technologyParis gold boxIn addition to holograms and security threads,paper money often includes special Guillochépatterns in the design to prevent counterfeiting.The sum and product of nested sinusoidalfunctions are used to form the blueprint of someof these patterns. Guilloché patterns have beencreated since the sixteenth century, but theirorigin is uncertain. They can be found carved inwooden door frames and etched on the metallicsurfaces of objects such as vases.We bLinkTo learnearn more aboutabGuilloché patterns, go to www.mcgrawhill.ca/school/learningcentres and follow the links.Investigate Expressions for sin (α β) and cos (α β)1. a) Draw a large rectangle and label its vertices A,AB, C, and D, where BC 2AB. Mark a point Eon BC. Join AE and use a protractor to draw EFperpendicular to AE. Label all right angles onyour diagram. Label BAE as α and EAF as β.βBαMaterials rulerE protractor1b) Measure the angles α and β. Use the angle sumof a triangle to determine the measures of allthe remaining acute angles in your diagram.Record their measures on the diagram.DFC2. a) Explain how you know that CEF α.b) Determine an expression for each of the other acute angles in thediagram in terms of α and β. Label each angle on your diagram.6.2 Sum, Difference, and Double-Angle Identities MHR 299

3. Suppose the hypotenuse AF of the inscribed right triangle has alength of 1 unit. Explain why the length of AE can be represented ascos β. Label AE as cos β.4. Determine expressions for line segments AB, BE, EF, CE, CF, AD, andDF in terms of sin α, cos α, sin β, and cos β. Label each side lengthon your diagram using these sines and cosines. Note that AD equalsthe sum of segments BE and EC, and DF equals AB minus CF.5. Which angle in the diagram is equivalent to α β? Determinepossible identities for sin (α β) and cos (α β) from ADF usingthe sum or difference of lengths. Compare your results with those ofa classmate.Reflect and Respond6. a) Verify your possible identities numerically using the measures of αand β from step 1. Compare your results with those of a classmate.b) Does each identity apply to angles that are obtuse? Are there anyrestrictions on the domain? Describe your findings.7. Consider the special case where α β. Write simplified equivalentexpressions for sin 2α and cos 2α.Link the IdeasIn the investigation, you discovered the angle sum identities for sine andcosine. These identities can be used to determine the angle sum identityfor tangent.The sum identities aresin (A B) sin A cos B cos A sin Bcos (A B) cos A cos B - sin A sin Btan A tan Btan (A B) 1 - tan A tan BThe angle sum identities for sine, cosine, and tangent can be used todetermine angle difference identities for sine, cosine, and tangent.We bLinkTo see a derivatderivationof the differencecos (A - B), go towww.mcgrawhill.ca/school/learningcentresand follow the links.300 MHR Chapter 6For sine,sin (A - B) sinsinsinsin(A (-B))A cos (-B) cos A sin (-B)A cos B cos A (-sin B)A cos B - cos A sin BThe three angle difference identities aresin (A - B) sin A cos B - cos A sin Bcos (A - B) cos A cos B sin A sin Btan A - tan Btan (A - B) 1 tan A tan BWhy is cos (-B) cos B?Why is sin (-B) -sin B?

A special case occurs in the angle sum identities when A B.Substituting B A results in the double-angle identities.For example, sin 2A sin (A A) sin A cos A cos A sin A 2 sin A cos ASimilarly, it can be shown thatcos 2A cos2 A - sin2 A2 tan Atan 2A 1 - tan2 AThe double-angle identities aresin 2A 2 sin A cos Acos 2A cos2 A - sin2 A2 tan Atan 2A 1 - tan2 AExample 1Simplify Expressions Using Sum, Difference, and Double-AngleIdentitiesWrite each expression as a single trigonometric function.a) sin 48 cos 17 - cos 48 sin 17 ππb) cos2 - sin233Solutiona) The expression sin 48 cos 17 - cos 48 sin 17 has thesame form as the right side of the difference identity for sine,sin (A - B) sin A cos B - cos A sin B.Thus,sin 48 cos 17 - cos 48 sin 17 sin (48 - 17 ) sin 31 ππb) The expression cos2 - sin2 has the same form as the right side of33the double-angle identity for cosine, cos 2A cos2 A - sin2 A.Therefore,π - sin2π cos 2πcos23332π cos3( ( ))How could you use technologyto verify these solutions?Your TurnWrite each expression as a single trigonometric function.a) cos 88 cos 35 sin 88 sin 35 ππb) 2 sin cos12126.2 Sum, Difference, and Double-Angle Identities MHR 301

Example 2Determine Alternative Forms of the Double-Angle Identity for CosineDetermine an identity for cos 2A that contains only the cosine ratio.SolutionAn identity for cos 2A is cos 2A cos2 A - sin2 A.Write an equivalent expression for the term containing sin A.Use the Pythagorean identity, cos2 A sin2 A 1.Substitute sin2 A 1 - cos2 A to obtain another form of the double-angleidentity for cosine.cos 2A cos2 A - sin2 Acos2 A - (1 - cos2 A)cos2 A - 1 cos2 A2 cos2 A - 1Your TurnDetermine an identity for cos 2A that contains only the sine ratio.Example 3Simplify Expressions Using Identities- cos 2x .Consider the expression 1sin 2xa) What are the permissible values for the expression?b) Simplify the expression to one of the three primary trigonometricfunctions.c) Verify your answer from part b), in the interval [0, 2π), usingtechnology.Solutiona) Identify any non-permissible values. The expression is undefinedwhen sin 2x 0.Method 1: Simplify the Double AngleUse the double-angle identity for sine to simplify sin 2x first.sin 2x 2 sin x cos x2 sin x cos x 0So, sin x 0 and cos x 0.sin x 0 when x πn, where n I.π πn, where n I.cos x 0 when x 2When these two sets of non-permissible values are combined, thepermissible values for the expression are all real numbers exceptπn , where n I.x 2302 MHR Chapter 6

Method 2: Horizontal Transformation of sin xFirst determine when sin x 0. Then, stretch the domain horizontally1.by a factor of2sin x 0 when x πn, where n I.πn , where n I.Therefore, sin 2x 0 when x 2- cos 2x are all realThe permissible values of the expression 1sin 2xπn , where n I.numbers except x 21 - (1 - 2 sin2 x)1 - cos 2xb) sin 2x2 sin x cos x2 sin2 x 2 sin x cos xsin x cosx tan xReplace sin 2x in the denominator.Replace cos 2x with the form of theidentity from Example 2 that willsimplify most fully.- cos 2x is equivalent to tan x.The expression 1sin 2x1 - cos 2xc) Use technology, with domain 0 x 2π, to graph y sin 2xand y tan x. The graphs look identical, which verifies, but does notprove, the answer in part b).Your Turnsin 2x .Consider the expressioncos 2x 1a) What are the permissible values for the expression?b) Simplify the expression to one of the three primary trigonometricfunctions.c) Verify your answer from part b), in the interval [0, 2π), usingtechnology.6.2 Sum, Difference, and Double-Angle Identities MHR 303

Example 4Determine Exact Trigonometric Values for AnglesDetermine the exact value for each expression.πa) sin12b) tan 105 Solutiona) Use the difference identity for sine with two special angles.π 3π -2π , useπ -π.For example, because41212126ππThe special angles and could alsoπ sinπ -π43sinbe used.4126Use sin (A - B)π cosπ - cosππ sin sin sin A cos B - cos A sin B.4 646 322 1 -222 2 6 2 -44 How could you verify this answer with62 a calculator?4()( )( ) ( )( )b) Method 1: Use the Difference Identity for TangentRewrite tan 105 as a difference of special angles.Are there other ways of writing 105 as thetan 105 tan (135 - 30 )sum or difference of two special angles?tan A - tan B .Use the tangent difference identity, tan (A - B) 1 tan A tan Btan 135 - tan 30 tan (135 - 30 ) 1 tan 135 tan 30 1-1 - 3 11 (-1) 31-1 - 3Simplify. 11- 31-1 - 3Multiply3- numerator and denominator by - 3 .1 31 - 3 How could you rationalize the3 1 denominator? 1- 3( )(304 MHR Chapter 6)()

Method 2: Use a Quotient Identity with Sine and Cosinesin 105 tan 105 cos 105 sin (60 45 )Use sum identities with special angles. Couldcos (60 45 )you use a difference ofsin 60 cos 45 cos 60 sin 45 angles identity here?cos 60 cos 45 - sin 60 sin 45 ( )( ) ( )( 12 )( 22 ) - ( 23 )( 22 ) 3 21 2 (2) 222 6 2 44 6 2-44 6 2 ( 6 2 2 - 64)(4 2- 6)How could you verify thatthis is the same answer as inMethod 1?Your TurnUse a sum or difference identity to find the exact values of11πa) cos 165 b) tan12Key IdeasYou can use the sum and difference identities to simplify expressions and todetermine exact trigonometric values for some angles.Sum IdentitiesDifference Identitiessin (A B) sin A cos B cos A sin Bsin (A - B) sin A cos B - cos A sin Bcos (A B) cos A cos B - sin A sin Btan A tan Btan (A B) 1 - tan A tan Bcos (A - B) cos A cos B sin A sin Btan A - tan Btan (A - B) 1 tan A tan BThe double-angle identities are special cases of the sum identities when thetwo angles are equal. The double-angle identity for cosine can be expressedin three forms using the Pythagorean identity, cos2 A sin2 A 1.Double-Angle Identitiessin 2A 2 sin A cos Acos 2A cos2 A - sin2 Acos 2A 2 cos2 A - 1cos 2A 1 - 2 sin2 A2 tan Atan 2A 1 - tan2 A6.2 Sum, Difference, and Double-Angle Identities MHR 305

Check Your UnderstandingPractise7. Simplify cos (90 - x) using a difference1. Write each expression as a singletrigonometric function.identity.8. Determine the exact value of eacha) cos 43 cos 27 - sin 43 sin 27 trigonometric expression.b) sin 15 cos 20 cos 15 sin 20 a) cos 75 b) tan 165 c) cos2 19 - sin2 19 7πc) sind) cos 195 3π5π5π3πd) sin cos - cos sin4422ππe) 8 sincos332. Simplify and then give an exact valuefor each expression.12πe) csc12(π12f) sin -)Applya) cos 40 cos 20 - sin 40 sin 20 b) sin 20 cos 25 cos 20 sin 25 ππc) cos2 - sin266ππππd) cos cos - sin sin23233. Using only one substitution, which formof the double-angle identity for cosinewill simplify the expression 1 - cos 2xto one term? Show how this happens.4. Write each expression as a singletrigonometric function.ππa) 2 sin cos44b) (6 cos2 24 - 6 sin2 24 ) tan 48 2 tan 76 c)1 - tan2 76 πd) 2 cos2 - 16πe) 1 - 2 cos2125. Simplify each expression to a singleprimary trigonometric function.sin 2θa)2 cos θb) cos 2x cos x sin 2x sin xcos 2θ 1c)2 cos θcos3 xd)cos 2x sin2 x6. Show using a counterexample thatthe following is not an identity:sin (x - y) sin x - sin y.306 MHR Chapter 6Yukon River at Whitehorse9. On the winter solstice, December 21 or 22,the power, P, in watts, received from thesun on each square metre of Earth can bedetermined using the equationP 1000 (sin x cos 113.5 cos x sin 113.5 ),where x is the latitude of the location inthe northern hemisphere.a) Use an identity to write the equation ina more useful form.b) Determine the amount of powerreceived at each location.i) Whitehorse, Yukon, at 60.7 Nii) Victoria, British Columbia, at 48.4 Niii) Igloolik, Nunavut, at 69.4 Nc) Explain the answer for part iii) above.At what latitude is the power receivedfrom the sun zero?

10. Simplify cos (π x) cos (π - x).11. Angle θ is in quadrant II and5 . Determine an exactsin θ 13value for each of the following.a) cos 2θ(value of sin 2x in terms of k?15. Show that each expression can besimplified to cos 2x.a) cos4 x - sin4 x2csc x - 2b)2b) sin 2θπc) sin θ 14. If (sin x cos x)2 k, then what is the)212. The double-angle identity for tangentin terms of the tangent function is2 tan x .tan 2x 1 - tan2 xa) Verify numerically that this equation isπ.true for x 6b) The expression tan 2x can also bewritten using the quotient identity forsin 2x . Verify thistangent: tan 2x cos 2xπ.equation numerically when x 6sin2xc) The expression from part b)cos 2x2 sin x cos xcan be expressed ascos2 x - sin2 xusing double-angle identities. Showhow the expression for tan 2x usedin part a) can also be rewritten in the2 sin x cos x .formcos2 x - sin2 x13. The horizontal distance, d, in metres,csc x16. Simplify each expression to the equivalentexpression shown.1 - cos 2xa)sin2 x24 - 8 sin2 x4b)tan 2x2 sin x cos x17. If the point (2, 5) lies on the terminal armof angle x in standard position, what is thevalue of cos (π x)?18. What value of k makes the equationsin 5x cos x cos 5x sin x 2 sin kx cos kxtrue?319. a) If cos θ and 0 θ 2π, determine5π .the value(s) of sin θ 63π2b) If sin θ - and θ 2π,32π .determine the value(s) of cos θ 3()(20. If A and B are both in quadrant I, and4 and cos B 12 , evaluate eachsin A 513of the following,travelled by a ball that is kicked at anangle, θ, with the ground is modelled by2(v0)2 sin θ cos θthe formula d , where v0gis the initial velocity of the ball, in metresper second, and g is the force of gravity(9.8 m/s2).Extenda) Rewrite the formula using a21. Determine the missing primarydouble-angle identity.b) Determine the angle θ (0 , 90 ) thatwould result in a maximum distancefor an initial velocity v0.c) Explain why it might be easier toanswer part b) with the double-angleversion of the formula that youdetermined in part a).)a) cos (A - B)b) sin (A B)c) cos 2Ad) sin 2Atrigonometric ratio that is required for the sin 2x to simplify toexpression2 - 2 cos2 xa) cos xb) 122. Use a double-angle identity for cosineto determine the half-angleformula for1 cosxx cosine, cos.22 6.2 Sum, Difference, and Double-Angle Identities MHR 307

23. a) Graph the curve y 4 sin x - 3 cos x.C2 a) Graph the function f(x) 6 sin x cos xNotice that it resembles a sine function.over the interval 0 x 360 .b) The function can be written as a sineb) What are the approximate valuesfunction in the form f(x) a sin bx.Compare how to determine this sinefunction from the graph versus usingthe double-angle identity for sine.of a and c for the curve in the formy a sin (x - c), where 0 c 90 ?c) Use the difference identity forsine to rewrite the curve fory 4 sin x - 3 cos x in the formy a sin (x - c).C3 a) Over the domain 0 x 360 , sketchthe graphs of y1 sin2 x and y2 cos2 x.How do these graphs compare?24. Write the following equation in the formy A sin Bx D, where A, B, and D areconstants:y 6 sin x cos3 x 6 sin3 x cos x - 3b) Predict what the graph of y1 y2 lookslike. Explain your prediction. Graph totest your prediction.c) Graph the difference of the twoCreate Connectionsfunctions: y1 - y2. Describe how the twofunctions interact with each other in thenew function.C1 a) Determine the value of sin 2x if3π using5 and π x cos x -132i) transformationsd) The new function from part c) isii) a double-angle identityb) Which method do you prefer? Explain.Project Cornersinusoidal. Determine the function inthe form f(x) a cos bx. Explain howyou determined the expression.Mach Numbers In aeronautics, the Mach number, M, of an aircraft is the ratio of its speed asit moves through air to the speed of sound in air. An aircraft breaks the soundbarrier when its speed is greater than the speed of sound in dry air at 20 C. When an aircraft exceeds Mach 1, M 1, a shockwave forms a cone that spreads backward and outward fromthe aircraft. The angle at the vertex of a cross-section of theθ.1 sincone is related to the Mach number by2M How could youuse the half-angle identity,1cos θ ,θsin 22to express the Mach number, M, as a function of θ? If plane A is travelling twice as fast as plane B,how are the angles of the cones formed by theplanes related?308 MHR Chapter 6θ

6.3Proving IdentitiesFocus on proving trigonometric identities algebraically understanding the difference between verifying and proving an identity showing that verifying that the two sides of a potential identity are equal for a givenvalue is insufficient to prove the identityMany formulas in science contain trigonometric functions. In physics,torque (τ), work (W ), and magnetic forces (FB) can be calculated usingthe following formulas:τ rF sin θW Fδr cos θFB qvB sin θIn dynamics, which is the branch of mechanics that d

b) To simplify the expression, use reciprocal and quotient identities to write trigonometric functions in terms of cosine and sine. cot x csc x cos x cos _x sin x _ 1 sin x cos x cos _x _sin x cos _x sin x 1 Your Turn a) Determine the non-permissible values, in radians, of the variable in th