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# Outline for MATH 1330 at UH

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This 6 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 16 views.

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Date Created: 02/06/15

Exercise Set 62 DoubleAngle and HalfAngle Formulas 12 Suppose that tanoi 73 and 37 lt 0 lt 27r Find a sin 20 b cos 20 c tan 20 Simplify each of the following expressions as much as possible without a calculator 13 14 15 16 17 18 19 20 21 22 23 25in15 cos15 cos2 sin2 2 2 zoos2 34quot71 17 25m2 12 2tan105 17 tan2 105 lisin2 x 72sin7 cos7 j 8 8 251n23 cos23 cos2 3 isinz 3 8 8 2tan j 12 litan2 11 12 25m2 7 71 12 Math 1330 Precai culus The University ofHaustan 24 25 26 27 28 sin2 675 7 cos2 675 72tan41 17 tan2 41 172cosz225 2cos2 71 2 2tan3oi tan2 3071 The formulas for sin and cos both contain a I sign meaning that a choice must be made as to whether or not the sign is positive or negative For each of the following examples rst state the quadrant in which the angle lies Then state whether the given expression is positive or negative Do not evaluate the expression 29 30 31 32 a cos105 b sin675quot a sin15quot b cos1125 a cos15 j b sin a sin b cos Chapter 6 Trigonometric F amulas and Equations Exercise Set 62 DoubleAngle and HalfAngle Formulas In the text tan is de ned as mm Sin S 2 1coss The following exercises can be used to derive this formula along with two additional formulas for tan 3 2 33 a b C 34 a b C Write the formula for sin Write the formula for cos Derive a new formula for tan using the Sin 6 identity tan 6 c 05 6 Leave both the numerator and denominator in radical form Show all work Where 65 2 This exercise will outline the derivation for s sin s tan In exercrse 33 it was 2 l cos s discovered that tan3 i 2 l cos s Rationalize the denominator by multiplying both the numerator and denominator by 1 coss Simplify the expression and write the result for tan j A detailed analysis of the signs of the trigonometric functions of s and in various quadrants reveals that the i symbol in part a is unnecessary This analysis is lengthy and will not be shown here Given this fact rewrite the formula from part a Without the i symbol How does this result from part b compare with the formula given in the text for tan 7 2 Math 1330 Precalculus The University ofHouStan 35 a b C 36 a b In exercise 33 it was discovered that tan i w 2 lcoss Rationalize the numerator by multiplying both the numerator and denominator by 17 coss Simplify the expression and write the new result for tanj A detailed analysis of the signs of the trigonometric functions of s and in various quadrants reveals that the i symbol in part a is unnecessary This analysis is lengthy and will not be shown here Given this fact rewrite the formula from part a Without the i symbol This gives yet another formula which can be used for tang Use the results from Exercises 3335 to write the three formulas for tan Which formula seems easiest to use and Why Which formula seems hardest to use and Why In the text tan j is defined as tang Sin S 2 lcoss Multiply both the numerator and denominator of the righthand side of the equation by 17 coss Then simplify to obtain a formula for tanj Show all work How does the result from part a compare to the identity obtained in part b of Exercise 39 Chapter 6 Trigonometric Formulas and Equations Exercise Set 62 DoubleAngle and HalfAngle Formulas Answer the following SHOW ALL WORK Do not leave any radicals in the denominator ie rationalize the denominator whenever appropriate 37 a Find cos75 by using a sum or difference 38 39 40 b C a b C a b C d a b C d formula Find cos75 by using a halfangle formula Enter the results from parts a and b into a calculator and round each one to the nearest hundredth Are they the same Find sin165 by using a sum or difference formula Find sin165 by using a halfangle formula Enter the results from parts a and b into a calculator and round each one to the nearest hundredth Are they the same Find sin1125 by using a halfangle formula Find cos1125 by using a halfangle formula Find tan1125 by computing s1n1 125 cos1125quot39 Find tan1125 by using a halfangle formula Find sin105 by using a halfangle formula Find cos105 by using a halfangle formula a sin 105 Find tan105 by computing W Find tan105 by using a halfangle formula Math 1330 Precalculus The University ofHaustan Find the exact value of each of the following by using a halfangle formula Do not leave any radicals in the denominator ie rationalize the denominator whenever appropriate 41 a b C 42 a b C 43 a b C 44 a b c 45 a b c 46 a b C 137 s1n 12 s1n1575 cos1575 tan1575 sin285 cos285 tan285 177 s1n 12 l77r COS 12 mpg 12 w Chapter 6 Trigonometric F amulas and Equations Exercise Set 62 DoubleAngle and HalfAngle Formulas MICD 1 Find the exact value of sin Answer the following MICD 47 If 0056 3 and 377 lt 6 lt 2 g Find the exact value of cos a lODfetgermine the quadrant of the terminal Side 01 Find the exact value of tan 3 I 6 b Complete the followmg 7 lt lt 7 2 J7 49 If tan67 and lt6lt7r c Determine the quadrant of the terminal Side 3 2 f 6 9 O 21 Find the exact value of sin3 d Based on the answer to part c determine 6 b Find the exactvalue of cos a the Sign of Sin 5 2 9 Based on the answer to Part 3 determine c Find the exact value of tan the sign of cos 5 37r 1 Find the exact value of sin 5039 If 0056 7 E and T lt 6 lt 2 6 21 Find the exact value of sin g Find the exact value of cos3 2 6 b Find the exact value of cos h Find the exact value of tan 2 c Find the exact value of tan J5 48 If sin6 7 and Ir lt6 lt3 5 2 Prove the following a Determine the quadrant of the terminal side Ofa 51 licos2x sin2x tanltxgt b Complete the following lt 3 lt 52 coslt3xgtmaniac2 cos x s1n x c Determine the quadrant of the terminal side Of 53 lixEsinxl 25inxcos2x d Based on the answer to part c determine 5439 054 x Sin4 x 0052x the sign of sin j 55 cscx 7 cotx tan e Based on the answer to part c determine the sign of cos Math 1330 Precalculus The University ofHaustan Chapter 6 Trigonometric Formulas and Equations Exercise Set 62 DoubleAngle and HalfAngle Formulas Math 1330 Precalculus The University afHaustan Chapter 6 Trigonomem39c Formulas and Equations Exercise Set 62 DoubleAngle and HalfAngle Formulas Answer the following 6 Derive the formula for tan26 by using a sum 1 2 Evaluate sin2 formula on tana 6 6 7 The sum formula for cosine yields the equation b Evaluate 25in cos26 cos2 67 sin2 6 To write cos26 strictly in terms of the sine function clssin2 2sinl 6 6 a Usmg the Pythagorean identity cos2 6sin2 6 l solve for cos2 6 b Substitute the result from part a into the above equation for cos 26 d Graph fx sin2x and gx 2sinx on the same set of axes e ls sin2x 2sinx 8 The sum formula for cosine yields the equation cos26cos2 67sin2 To write Evaluate cos 2 cos26 strrctly in terms of the cosrne function 21 Using the Pythagorean identity cos2 6sin2 6 l solve for sin2 6 c IS COS 2 1 Zoos 1 7 b Substitute the result from part a into the 6 6 above equation for cos26 b Evaluate 2cos j d Graph fx cos2x and g x 2cosx on the same set of axes Answer the fonowmg e Is cosx 2cosx 9 Suppose that cos0 and 3T lt 0 lt 27r Find 3 2 Evaluate tan2 a smaa b cos20 b Evaluate 2tan j c tansz It II C IS tan2g 2tan gjrf 10 Suppose that tanzx and Ir lt or lt37 Find d Graph fx tan2x and a sin20 gx 2tanx on the same set of axes b 2 e ls tan2x 2tanx cos a c tansz 4 Derive the formula for sin26 by using a sum 2 7r formula on Sinw g 11 Suppose that s1no7g and 3lt 0 lt 7 Find 5 Derive the formula for cos26 by using a sum 2 sin2a formula on cos66 b COSQD c tan 20 Math 1330 Precalculus The University ofHaustan Chapter 6 Trigonometric Formulas and Equations

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