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# Precalculus MATH 1330

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This 6 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 1330 at University of Houston taught by Staff in Fall. Since its upload, it has received 50 views. For similar materials see /class/208417/math-1330-university-of-houston in Mathmatics at University of Houston.

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Date Created: 09/19/15

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 I 2 7r 2 7r 7 S sm E 7 sm E 39 21 Using the Pythagorean identity 2 2 7 2 Graph fx sinzx and gltxgt Zsinx cos 6s1n 6 7 l solve for cos on the same set of axes b Substitute the result from part a into the above equation for cos26 A n V A O V A a V 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 9 A 5 V 21 Using the Pythagorean identity cos2 6sin2 6 l solve for sin2 6 IS COS 2 1 Zoos 1 7 b Substitute the result from part a into the 6 above equation for cos26 A 5quot V Evaluate 2 cos A n V d Graph fx cos2x and g x 2cosx on the same set of axes Answer the fouowmg39 e Is cosx 2cosx 9 Suppose that coszx and 3T lt 0 lt 27r Find 3 2 Evaluate tan2 a smaa b cos2a b Evaluate 2tanj c tan2a It II C IS tan2g 2tang 10 Suppose that tanzx and Ir lt or lt37 d Graph fx tan2x and Fmd gx 2tanx on the same set of axes a gnaw b 2 e ls tan2x 2tanx cos a c tan2a 4 Derive the formula for sin26 by using a sum 2 formula on sing g 11 Suppose that s1no g and 3 lt 0 lt 7 Find 5 Derive the formula for cos26 by using a sum 2 sinsz formula on cos66 b cosza c tan 20 Math 1330 Precalculus The University quousfon Chapter 6 T gononzemc Formulas and Equations Exercise Set 62 DoubleAngle and HalfAngle Formulas 12 Suppose that tanzx 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 p A UI H P p A l H gt5 N W 2sin15 cos15 cos2 sin2 2 2 zoos2 34quot71 17 2sin2 12 2tan105 17mm2 105 lisin2 x 7n 7 725m cos 8 8 2sin23 cos23 cos2 3 isinz 3 8 8 2tan j 12 17 tan2 11 12 2sin2 7 71 12 Math 1330 Precalculus The University quausfon 24 sin2 675 7 cos2 675 72tan41 39 litan241 N UI N a 172cosz225 27 2cos2371 2 28 2tan3o 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 a cos105 b sin675quot 30 a sin15quot cos1125 A 5quot V 31 a cos15 j b sin 32 a 51117 Chapter 6 Trigonomerric F 0mmlas and Equations Exercise Set 62 DoubleAngle and HalfAngle Formulas In the text tan is de ned as tan 3 2 W W sins tang 1coss The following exercises can be used to derive this formula along with two additional formulas for a b C A 5quot V Write the formula for sin Write the formula for cos Derive a new formula for tan using the 9 identity tan 9 m Where 6 5 0056 2 Leave both the numerator and denominator in radical form Show all work This exercise will outline the derivation for sins tan 3 In exercise 33 it was 2 1 cos discovered that tan3 i w 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 quausfon 35 a A n V 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 s tan 2 Use the results from Exercises 3335 to 2 formula seems easiest to use and Why Which formula seems hardest to use and Why write the three formulas for tan j Which In the text tan j is defined as s sin s tan 2 l cos s 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 T gonomemc F ommlas and Equations Exercise Set 62 DoubleAngle and HalfAngle Formulas Answer the following SHOW ALL WORK Do not Find the exact value of each of the following by using a leave any radicals in the denominator ie rationalize halfangle formula Do not leave any radicals in the the denominator whenever appropriate denominator ie rationalize the denominator whenever appropriate 37 a Find cos75 by using a sum or difference 137r formula 41 3 Sln j b Find cos75 by using a halfangle 13 b cos formula 12 c Enter the results from parts a and b into a t 137r calculator and round each one to the nearest c an E hundredth Are they the same 38 a Find sin165 by using a sum or difference 3 42 a Sln formula 8 b Find sin165 by using a halfangle 1 cos 8 formula 3 c Enter the results from parts a and b into a c tan calculator and round each one to the nearest 8 hundredth Are they the same 43 117 39 a Find sin1125 by using a halfangle 239 sm Y f 1 ormu a b cos b Find cos1125 by using a halfangle 8 1 17 formula c tanj c Find tan1125 by computing sin 1 125 44 a sin1575 cos 1 125 A O V Find tan1125 by using a halfangle formula 40 a Find sin105 by using a halfangle 45 a Sin285 formula b Find cos105 by using a halfangle formula sin105 c Find tan105 by computing cos105 177 46 a Sln E d Find tan105 by using a halfangle 17 It formula b cos 177r c tan lt 12 Math 1330 Precalculus The University quausfon Chapter 6 T gononzemc Formulas and Equations Exercise Set 62 DoubleAngle and HalfAngle Formulas 1 Find the exact value of sin w Answer the following 47 If 0056 2 and 3 7 lt 6 lt 2 g Find the exact value of cos m MICD w a lODfetgermine the quadrant of the terminal Side 01 Find the exact value of tan 3 w b Complete the following lt a lt 2 J7 7r 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 exact value of cos the Sign of Sin 5 2 e Based on the answer to part c 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 w 21 Find the exact value of sin g Find the exact value of cos 2 m MICD b Find the exact value of cosj h Find the exact value of tan w c Find the exact value of tan m 37r lf sin67 and 7rlt6lt 5 2 5 W Prove the following a Determine the quadrant of the terminal side of 6 l 7 cos 2x 51 tanx b Complete the following sm2x lt 2 lt 52 cos3x7sin3x 72 cosx s1nx c Determine the quadrant of the terminal side Of 53 lixEsinxl 25inxcos2x 2 d Based on the answer to part c determine 5439 054 x Sin4 x 0052x the Slgn Of 5 55 cscx 7 cotx tan e Based on the answer to part c determine the sign of cos Math 1330 Precalculus The University quousfon Chapter 6 Trigonomen39ic Formulas and Equations Exercise Set 62 DoubleAngle and HalfAngle Formulas 1 tan2 f 56 2 1 7 tan2 5 2 sec6 WW Math 1330 Precalculus The University QfHausfan Chapter 6 Trigonomem39c Formulas and Equations

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