7-2: Verifying Trigonometric Identities

While working on amathematics assignment, a group of studentsp li c a tiderived an expression for the length of a ladderthat, when held horizontally, would turn from a 5-foot widecorridor into a 7-foot wide corridor. They determined that themaximum length of a ladder that would fit was given byPROBLEM SOLVINGonAp Use the basictrigonometricidentities toverify otheridentities. Find numericalvalues oftrigonometricfunctions.l WorealdOBJECTIVESVerifying TrigonometricIdentitiesR7-27 ft 7 sin 5 cos sin cos ( ) , where is the angle that the laddermakes with the outer wall of the 5-foot wide corridor. Whentheir teacher worked the problem, she concluded that ( ) 7 sec 5 csc . Are the two expressions for ( ) equivalent? This problem will be solved in Example 2.5 ftVerifying trigonometric identities algebraically involves transforming oneside of the equation into the same form as the other side by using the basictrigonometric identities and the properties of algebra. Either side may betransformed into the other side, or both sides may be transformed separatelyinto forms that are the same.Suggestionsfor VerifyingTrigonometricIdentities Transform the more complicated side of the equation into the simpler side. Substitute one or more basic trigonometric identities to simplifyexpressions. Factor or multiply to simplify expressions. Multiply expressions by an expression equal to 1. Express all trigonometric functions in terms of sine and cosine.You cannot add or subtract quantities from each side of an unverified identity,nor can you perform any other operation on each side, as you often do withequations. An unverified identity is not an equation, so the properties of equalitydo not apply.Example1 Verify that sec2 x tan x cot x tan2 x is an identity.Since the left side is more complicated, transform it into the expression on theright.sec2 x tan x cot x tan2 x1tan xsec2 x tan x tan2 xsec2 x 1 tan2 xx 1 1 tan2 xtan2 x tan2 xtan21tan xcot x Multiply.sec2 x tan2 x 1Simplify.We have transformed the left side into the right side. The identity is verified.Lesson 7-2Verifying Trigonometric Identities431

r7 sin 5 cos sin cos that 7 sec 5 csc is an identity.AponeldRExamples 2 PROBLEM SOLVING Verify that the two expressions for ( ) in theapplication at the beginning of the lesson are equivalent. That is, verifyal Wop li c a tiBegin by writing the right side in terms of sine and cosine.7 sin 5 cos 7 sec 5 csc sin cos 7 sin 5 cos 75 sin cos cos sin sec , csc 7 sin 5 cos 7 sin 5 cos sin cos sin cos sin cos Find a common denominator.7 sin 5 cos 7 sin 5 cos sin cos sin cos Simplify.1cos 1sin The students and the teacher derived equivalent expressions for ( ), thelength of the ladder.sin Acos A csc2 A cot2 A is an identity.3 Verify that csc Asec ASince the two sides are equally complicated, we will transform each sideindependently into the same form.sin Acos A csc2 A cot2 Acsc Asec A22cos Asin A (1 cot A) cot A11 cos Asin Asin2 A cos2 A 1Quotient identities;Pythagorean identitySimplify.1 1sin2 A cos2 A 1The techniques that you use to verify trigonometric identities can also beused to simplify trigonometric equations. Sometimes you can change an equationinto an equivalent equation involving a single trigonometric function.Examplecot x 2.4 Find a numerical value of one trigonometric function of x if cos xYou can simplify the trigonometric expression on the left side by writing it interms of sine and cosine.cot x 2cos xcos x sin x 2cos xcos x1 2sin x cos x432Chapter 7cos xcot x sin xDefinition of divisionTrigonometric Identities and Equations

1 2sin xSimplify.1sin xcsc x 2 csc xcot xcos xTherefore, if 2, then csc x 2.You can use a graphing calculator to investigate whether an equation may bean identity.GRAPHING CALCULATOR EXPLORATION Graph both sides of the equation as twoseparate functions. For example, to testsin2 x (1 cos x)(1 cos x), graphy1 sin2 x and y2 (1 cos x)(1 cos x)on the same screen. If the two graphs do not match, then theTRY THESE Determine whether eachequation could be an identity. Write yes or no.1. sin x csc x sin2 x cos2 x2. sec x csc x 11csc x sec x3. sin x cos x equation is not an identity. If the two sides appear to match in everywindow you try, then the equation may bean identity.WHAT DO YOU THINK?4. If the two sides appear to match in everywindow you try, does that prove that theequation is an identity? Justify youranswer.sec x cos x5. Graph the function f(x) .tan xWhat simpler function could you set equalto f(x) in order to obtain an identity?[ , ] scl:1 by [ 1, 1] scl:1C HECKCommunicatingMathematicsFORU N D E R S TA N D I N GRead and study the lesson to answer each question.1. Write a trigonometric equation that is not an identity. Explain how you know it isnot an identity.2. Explain why you cannot square each side of the equation when verifying atrigonometric identity.3. Discuss why both sides of a trigonometric identity are often rewritten in terms ofsine and cosine.Lesson 7-2 Verifying Trigonometric Identities433

4. MathJournal Create your own trigonometric identity that contains at leastthree different trigonometric functions. Explain how you created it. Give it toone of your classmates to verify. Compare and contrast your classmate’sapproach with your approach.Guided PracticeVerify that each equation is an identity.cot x5. cos x csc x1cos x6. tan x sec xsin x 117. csc cot csc cot 8. sin tan sec cos 9. (sin A cos A)2 1 2 sin2 A cot AFind a numerical value of one trigonometric function of x.110. tan x sec x411. cot x sin x cos x cot x12. OpticsThe amount of light thata source provides to a surfaceis called the illuminance. Theilluminance E in foot candles ona surface that is R feet from asource of light with intensityPerpendicularto surfaceII cos RI candelas is E 2 , where REis the measure of the anglebetween the direction of the lightand a line perpendicular to the surfacebeing illuminated. Verify thatI cot R csc E 2 is an equivalent formula.E XERCISESPracticeVerify that each equation is an identity.ABsec A13. tan A csc A14. cos sin cot 1 sin x15. sec x tan x cos x1 tan x16. sec xsin x cos x2 sin2 118. sin cos sin cos 17. sec x csc x tan x cot x2 sec A csc A19. (sin A cos A)2 sec A csc Acos y1 sin y21. 1 sin ycos ycot2 x23. csc x 1 csc x 125. sin cos tan cos2 120. (sin 1)(tan sec ) cos 22. cos cos ( ) sin sin ( ) 124. cos B cot B csc B sin B1 cos x26. (csc x cot x)2 1 cos xcos xsin x27. sin x cos x 1 tan x1 cot x28. Show that sin cos tan sin sec cos tan .434Chapter 7 Trigonometric Identities and Equationswww.amc.glencoe.com/self check quiz

Find a numerical value of one trigonometric function of x.Ccsc x29. 2 cot x1 tan x30. 21 cot x1sec x31. cos xcot xcsc xsin x1 cos x32. 41 cos xsin x33. cos2 x 2 sin x 2 034. csc x sin x tan x cos xtan3 135. If sec2 1 0, find cot .tan 1GraphingCalculatorUse a graphing calculator to determine whether each equation could be anidentity.1136. 1sin2 xcos2 x37. cos (cos sec ) sin2 sin3 x cos3 x38. 2 sin A (1 sin A)2 2 cos2 A 39. sin2 x cos2 xsin x cos x40. Electronicsa. Write an expression for the power in terms of cos2 2 ft.onWhen an alternating current of frequency f and peak current I0passes through a resistance R, then the power delivered to the resistance attime t seconds is P I02R sin2 2 ft.ldRApplicationsand ProblemSolvingb. Write an expression for the power in terms of csc2 2 ft.l WoreaApp li c a ti41. Critical Thinking12 2 2xLet x tan where . Write f(x) 4x2 1 in terms of a single trigonometric function of .42. Spherical Geometry is the Greekletter beta and is the Greekletter gamma.Spherical geometry is thegeometry that takes place on the surface of asphere. A line segment on the surface of the sphereis measured by the angle it subtends at the centerof the sphere. Let a, b, and c be the sides of a righttriangle on the surface of the sphere. Let the anglesopposite those sides be , , and 90 ,respectively. The following equations are true:caa cbbsin a sin sin ccos sin cos b cos c cos a cos b.Show that cos tan a cot c.43. PhysicsWhen a projectile is fired from the ground, its height y and horizontal gx 22v0 cos x sin cos displacement x are related by the equation y , where v0 is2 2the initial velocity of the projectile, is the angle at which it was fired, and gis the acceleration due to gravity. Rewrite this equation so that tan is theonly trigonometric function that appears in the equation.Lesson 7-2 Verifying Trigonometric Identities435

Consider a circle O with radius 1. PA and T B are each perpendicular to O B . Determine the areaof ABTP as a product of trigonometric functions of .T44. Critical ThinkingP OABLet a, b, and c be the sides of a triangle. Let , , and be therespective opposite angles. Show that the area A of the triangle is given by45. Geometrya2 sin sin 2 sin ( )A .Mixed Reviewtan x cos x sin x tan x46. Simplify . (Lesson 7-1)sec x tan x.47. Write an equation of a sine function with amplitude 2, period 180 , and phaseshift 45 . (Lesson 6-5)15 48. Change radians to degree measure to the nearest minute. (Lesson 6-1)1649. Solve 3y 1 2 0. (Lesson 4-7)350. Determine the equations of the vertical and horizontal asymptotes, if any, of3xf(x) . (Lesson 3-7)x 151. ManufacturingThe Simply Sweats Corporation makes high qualitysweatpants and sweatshirts. Each garment passes through the cuttingand sewing departments of the factory. The cutting and sewing departmentshave 100 and 180 worker-hours available each week, respectively. The fabricsupplier can provide 195 yards of fabric each week. The hours of work andyards of fabric required for each garment are shown in the table below. Ifthe profit from a sweatshirt is 5.00 and the profit from a pair of sweatpantsis 4.50, how many of each should the company make for maximum profit?(Lesson 2-7)Simply Sweats Corporation“Quality Sweatpants and 5 h1.5 ydPants1.5 h2h3 yd52. State the domain and range of the relation {(16, 4),(16, 4)}. Is this relation afunction? Explain. (Lesson 1-1)53. SAT/ACT PracticeA1D 1436a bb aa bb a(a b)2B (a b)2Divide by .Chapter 7 Trigonometric Identities and Equations1 C a2 b2E 0Extra Practice See p. A38.

Use the basic trigonometric identities to verify other identities. Find numerical values of trigonometric functions. 7 ft 5 ft Transform the more complicated side of the equation into the simpler side. Substitute one or more basic trigonometric identities to simplify expressions. Factor or multiply to simplify expressions.