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# FINAL STUDY GUIDE MAC1114

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This 65 page Study Guide was uploaded by mak15k on Sunday December 6, 2015. The Study Guide belongs to MAC1114 at Florida State University taught by Dr. LeNoir in Fall 2015. Since its upload, it has received 83 views. For similar materials see Analytic Trigonometry MAC1114 in Math at Florida State University.

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Date Created: 12/06/15

MAC1114 LeNoir STUDY GUIDE FOR FINAL THIS FINAL IS CUMULATIVE SO BASICALLY THIS STUDY GUIDE IS GOING TO INCLUDE ALL MY NOTES FROM THIS YEAR SINCE THE ONLY WAY TO REALLY STUDY IS TO PRACTICE AND MY NOTES WILL SHOW YOU HOW TO DO THAT ALSO INCLUDED IN THIS STUDY GUIDE ARE HINTFORMULA SHEETS AND EXAMPLES OF WORKED PROBLEMS NOTEBOOK PAPER FROM DR LENOIR HERSELF AS I39VE SAID BEFORE THE ONLY WAY TO TRULY GET TRIG DOWN IS TO PRACTICE SO BUCKLE DOWN AND START STUDYING CHAPTERS 5156 Attached notes and pictures with examples CHAPTERS 63 66 67 7173 81 82 63 DOUBLE AND HALF ANGLES see formulas for 63 1 write down the memorized identity that you need to use to solve the question 2 is the value gt locate the quadrant Quadrantl All P39ositive QuadrantZ Sin and Cse P39ositive Quadrant Tan and Cot P39ositive Quadrant l Cosano See Positive MAC1114 LeNoir 3 multiply or divide original interval by trig constant to get other interval 4 use quadrant of common value of steps 2 amp 3 5 draw trig function in correct quadrant 6 nd trig value within formula identity from picture 7 solveis there a square root find out if it39s trigewhich interval belongs to 6 see examples for 63 66 SOLVING TRIG EQUATIONS To solve means to find all possible values of 9 within the given interval 1 get the reference angle gt locate the quadrant gt draw the anges in the correct quadrant 2 based off where these drawn angles are name them accordingly 3 set the trig value of Gequal to these angles and solve for e if the problem only asks for general solutions stop here 4 add and subtract the period of the trig function to find all values of 9 within the given interval these are the speci c solutions To nd the number of solutions 1 don39t get distracted by the value of the trig function 2 nd the location of the terminal side gt locate the quadrant gt draw the anges in the correct quadrant 3 draw the interval and mark the angles that are within this interval pay attention to the signs see examples for 66 67 SOLVING TRIG EQUATIONS MAC1114 LeNoir see hints for 67 set to zero and factoruse an identity rst then factormultiply both sides by reciprocal then continuewrite in terms of sin or cos see examples for 67 7173 SOLVING TRIANGLES see hints for 7173 To solve a triangle means to nd all the lengths of all three sides and the measures of all three angles Angle of elevation the acute angle formed by a horizontal line and an observer39s line of sight to an object above the horizontal Angle of depression the acute angle formed by a horizontal line and an observer39s line of sight to an object below the horizontal Bearing acute angle formed between the line segment and the northsouth line through the point see examples for 7173 MAC1114 LeNoir STUDY GUIDE FOR FINAL THIS FINAL IS CUMULATIVE SO BASICALLY THIS STUDY GUIDE IS GOING TO INCLUDE ALL MY NOTES FROM THIS YEAR SINCE THE ONLY WAY TO REALLY STUDY IS TO PRACTICE AND MY NOTES WILL SHOW YOU HOW TO DO THAT ALSO INCLUDED IN THIS STUDY GUIDE ARE HINTFORMULA SHEETS AND EXAMPLES OF WORKED PROBLEMS NOTEBOOK PAPER FROM DR LENOIR HERSELF AS I39VE SAID BEFORE THE ONLY WAY TO TRULY GET TRIG DOWN IS TO PRACTICE SO BUCKLE DOWN AND START STUDYING CHAPTERS 5156 Attached notes and pictures with examples CHAPTERS 63 66 67 7173 81 82 63 DOUBLE AND HALF ANGLES see formulas for 63 1 write down the memorized identity that you need to use to solve the question 2 is the value gt locate the quadrant Quadrantl All P39ositive QuadrantZ Sin and Cse P39ositive Quadrant Tan and Cot P39ositive Quadrant l Cosano See Positive MAC1114 LeNoir 3 multiply or divide original interval by trig constant to get other interval 4 use quadrant of common value of steps 2 amp 3 5 draw trig function in correct quadrant 6 nd trig value within formula identity from picture 7 solveis there a square root find out if it39s trigewhich interval belongs to 6 see examples for 63 66 SOLVING TRIG EQUATIONS To solve means to find all possible values of 9 within the given interval 1 get the reference angle gt locate the quadrant gt draw the anges in the correct quadrant 2 based off where these drawn angles are name them accordingly 3 set the trig value of Gequal to these angles and solve for e if the problem only asks for general solutions stop here 4 add and subtract the period of the trig function to find all values of 9 within the given interval these are the speci c solutions To nd the number of solutions 1 don39t get distracted by the value of the trig function 2 nd the location of the terminal side gt locate the quadrant gt draw the anges in the correct quadrant 3 draw the interval and mark the angles that are within this interval pay attention to the signs see examples for 66 67 SOLVING TRIG EQUATIONS MAC1114 LeNoir see hints for 67 set to zero and factoruse an identity rst then factormultiply both sides by reciprocal then continuewrite in terms of sin or cos see examples for 67 7173 SOLVING TRIANGLES see hints for 7173 To solve a triangle means to nd all the lengths of all three sides and the measures of all three angles Angle of elevation the acute angle formed by a horizontal line and an observer39s line of sight to an object above the horizontal Angle of depression the acute angle formed by a horizontal line and an observer39s line of sight to an object below the horizontal Bearing acute angle formed between the line segment and the northsouth line through the point see examples for 7173 MAC1114 LeNoir STUDY GUIDE FOR FINAL THIS FINAL IS CUMULATIVE SO BASICALLY THIS STUDY GUIDE IS GOING TO INCLUDE ALL MY NOTES FROM THIS YEAR SINCE THE ONLY WAY TO REALLY STUDY IS TO PRACTICE AND MY NOTES WILL SHOW YOU HOW TO DO THAT ALSO INCLUDED IN THIS STUDY GUIDE ARE HINTFORMULA SHEETS AND EXAMPLES OF WORKED PROBLEMS NOTEBOOK PAPER FROM DR LENOIR HERSELF AS I39VE SAID BEFORE THE ONLY WAY TO TRULY GET TRIG DOWN IS TO PRACTICE SO BUCKLE DOWN AND START STUDYING CHAPTERS 5156 Attached notes and pictures with examples CHAPTERS 63 66 67 7173 81 82 63 DOUBLE AND HALF ANGLES see formulas for 63 1 write down the memorized identity that you need to use to solve the question 2 is the value gt locate the quadrant Quadrantl All P39ositive QuadrantZ Sin and Cse P39ositive Quadrant Tan and Cot P39ositive Quadrant l Cosano See Positive MAC1114 LeNoir 3 multiply or divide original interval by trig constant to get other interval 4 use quadrant of common value of steps 2 amp 3 5 draw trig function in correct quadrant 6 nd trig value within formula identity from picture 7 solveis there a square root find out if it39s trigewhich interval belongs to 6 see examples for 63 66 SOLVING TRIG EQUATIONS To solve means to find all possible values of 9 within the given interval 1 get the reference angle gt locate the quadrant gt draw the anges in the correct quadrant 2 based off where these drawn angles are name them accordingly 3 set the trig value of Gequal to these angles and solve for e if the problem only asks for general solutions stop here 4 add and subtract the period of the trig function to find all values of 9 within the given interval these are the speci c solutions To nd the number of solutions 1 don39t get distracted by the value of the trig function 2 nd the location of the terminal side gt locate the quadrant gt draw the anges in the correct quadrant 3 draw the interval and mark the angles that are within this interval pay attention to the signs see examples for 66 67 SOLVING TRIG EQUATIONS MAC1114 LeNoir see hints for 67 set to zero and factoruse an identity rst then factormultiply both sides by reciprocal then continuewrite in terms of sin or cos see examples for 67 7173 SOLVING TRIANGLES see hints for 7173 To solve a triangle means to nd all the lengths of all three sides and the measures of all three angles Angle of elevation the acute angle formed by a horizontal line and an observer39s line of sight to an object above the horizontal Angle of depression the acute angle formed by a horizontal line and an observer39s line of sight to an object below the horizontal Bearing acute angle formed between the line segment and the northsouth line through the point see examples for 7173 MAC1114 Molly Kitchen THURSDAY 08272015 LECTURE NOTES 51 go to the Einstein website to practice what was covered Help session start next week Quiz 1 on Sept 2nd practice on Einstein website SECTION 51 An angle should be thought of as motion start vs stop position determines measurement InItIal Side counterclockWIse motionstoppng pOSItlon IS term de positive angle InItIal Side clockmse motlon gt negative angle RIGHT ANGLE 90 STRAIGHT ANGLE 180 4 0ltthetalt90 is an acute angle Initial side is always positive xaxis terminal is still where you stop rotating coordinate plane 0 angle no movement 90 180 270 360 one complete revolution CCW 90 180 270 360 negative meaning CW movement Quadrantal angles the terminal angle positions that lie on the axes 1 1360 of revolution so 1 revolution is 360 2n n180 radian 1 1 radian 180n 1 Convert 150 to radians 150 x n180 5n6 radians reductions per numerator denominator 2 Convert the angle 34 radians to degrees if no degree smbol not radians 34 X 18011 l35T MAC1114 Molly Kitchen Theorem For a circle of radius 39r39 a central angle of theta radians subtends an arc whose length is 39s39 S Also the area of the sector is given by A MEMORIZE THESE FORMULAS PROBEM SET 1 1 Draw a picture 5 re 16 4 e 4 e 2 A l2r2theta A 12 32 RADIANS 30 x n180 n6 A 12 32 Tl6 A 3114 sq ft 3 60 if string length is 30 inches 5 re s 30T3 s lOn 4 36inch long windshield wipersl3 a revolution 5 r6 theta 13 revolution 5 36 Zn3 theta 13 Zn3 5 95 arc 95 angle 95 x n180 19n 36 s re OR 95n180 95 r95r180 1809511 x 95 r 95n180 x 1809511 Linear speed amp angular speed MAC1114 Molly Kitchen Linear distance time Angular angle time since sre st retst r x et PROBLEM SET 2 l v dt ft per second 2 w et 2040 3 v rw 18w w in angular speed 33 13 revolutions per minute 1003 rpm 1003 x 211 200n3 RADIANS per minute so v 18 x 200n3 4vrw2x3l MAC1114 Molly Kitchen THURSDAY 08272015 LECTURE NOTES 51 go to the Einstein website to practice what was covered Help session start next week Quiz 1 on Sept 2nd practice on Einstein website SECTION 51 An angle should be thought of as motion start vs stop position determines measurement InItIal Side counterclockWIse motionstoppng pOSItlon IS term de positive angle InItIal Side clockmse motlon gt negative angle RIGHT ANGLE 90 STRAIGHT ANGLE 180 4 0ltthetalt90 is an acute angle Initial side is always positive xaxis terminal is still where you stop rotating coordinate plane 0 angle no movement 90 180 270 360 one complete revolution CCW 90 180 270 360 negative meaning CW movement Quadrantal angles the terminal angle positions that lie on the axes 1 1360 of revolution so 1 revolution is 360 2n n180 radian 1 1 radian 180n 1 Convert 150 to radians 150 x n180 5n6 radians reductions per numerator denominator 2 Convert the angle 34 radians to degrees if no degree smbol not radians 34 X 18011 l35T MAC1114 Molly Kitchen Theorem For a circle of radius 39r39 a central angle of theta radians subtends an arc whose length is 39s39 S Also the area of the sector is given by A MEMORIZE THESE FORMULAS PROBEM SET 1 1 Draw a picture 5 re 16 4 e 4 e 2 A l2r2theta A 12 32 RADIANS 30 x n180 n6 A 12 32 Tl6 A 3114 sq ft 3 60 if string length is 30 inches 5 re s 30T3 s lOn 4 36inch long windshield wipersl3 a revolution 5 r6 theta 13 revolution 5 36 Zn3 theta 13 Zn3 5 95 arc 95 angle 95 x n180 19n 36 s re OR 95n180 95 r95r180 1809511 x 95 r 95n180 x 1809511 Linear speed amp angular speed MAC1114 Molly Kitchen Linear distance time Angular angle time since sre st retst r x et PROBLEM SET 2 l v dt ft per second 2 w et 2040 3 v rw 18w w in angular speed 33 13 revolutions per minute 1003 rpm 1003 x 211 200n3 RADIANS per minute so v 18 x 200n3 4vrw2x3l MAC1114 Molly Kitchen THURSDAY 08272015 LECTURE NOTES 51 go to the Einstein website to practice what was covered Help session start next week Quiz 1 on Sept 2nd practice on Einstein website SECTION 51 An angle should be thought of as motion start vs stop position determines measurement InItIal Side counterclockWIse motionstoppng pOSItlon IS term de positive angle InItIal Side clockmse motlon gt negative angle RIGHT ANGLE 90 STRAIGHT ANGLE 180 4 0ltthetalt90 is an acute angle Initial side is always positive xaxis terminal is still where you stop rotating coordinate plane 0 angle no movement 90 180 270 360 one complete revolution CCW 90 180 270 360 negative meaning CW movement Quadrantal angles the terminal angle positions that lie on the axes 1 1360 of revolution so 1 revolution is 360 2n n180 radian 1 1 radian 180n 1 Convert 150 to radians 150 x n180 5n6 radians reductions per numerator denominator 2 Convert the angle 34 radians to degrees if no degree smbol not radians 34 X 18011 l35T MAC1114 Molly Kitchen Theorem For a circle of radius 39r39 a central angle of theta radians subtends an arc whose length is 39s39 S Also the area of the sector is given by A MEMORIZE THESE FORMULAS PROBEM SET 1 1 Draw a picture 5 re 16 4 e 4 e 2 A l2r2theta A 12 32 RADIANS 30 x n180 n6 A 12 32 Tl6 A 3114 sq ft 3 60 if string length is 30 inches 5 re s 30T3 s lOn 4 36inch long windshield wipersl3 a revolution 5 r6 theta 13 revolution 5 36 Zn3 theta 13 Zn3 5 95 arc 95 angle 95 x n180 19n 36 s re OR 95n180 95 r95r180 1809511 x 95 r 95n180 x 1809511 Linear speed amp angular speed MAC1114 Molly Kitchen Linear distance time Angular angle time since sre st retst r x et PROBLEM SET 2 l v dt ft per second 2 w et 2040 3 v rw 18w w in angular speed 33 13 revolutions per minute 1003 rpm 1003 x 211 200n3 RADIANS per minute so v 18 x 200n3 4vrw2x3l PARTIAL NOTES for 61 62 Section 61 Recall the following identities BASIC IDENTITIES PYTHAGOREAN IDENTITIES I 2 2 tang 8mg CSC9 s1n 9cos 91 cos6 s1n6 2 2 cou9cfm9 36mg tan 91sec 9 s1n0 c039 2 2 cote 1cot H csc 0 tan6 EVENODD IDENTITIES sin 0 sinB csc 6 csc6 cos 9 cosB sec 6 secH tan 6 tan9 cot 0 cot 9 IDENTITIES FOR REDUCING FUNCTIONS sin 9 c039 cos 3 9 sin6 tan 3 9 cot 9 sin 6 cosH cos 6 sin6 tan 6 cot 6 sin2ar 9 sin9 cos2ar 9 cosH tan2ar 9 tan9 sin2ar 9 sinH cos2ar 9 cosH tan2ar 9 tan9 sinar 9 sin9 COSJ397 9 cosH tanat 9 tan9 sinar 9 sin9 cosar 9 cos 6 tanat 9 tanH sin 9 cos9 cos 9 sin9 tan37 9 cot 9 sin37 6 cos9 cos37 9 sin6 tan37 6 cot 6 Must also be able to recognize variations for example sin2 9 1 cos2 9 sin6 csc 6 TO ESTABLISH IDENTITIES I start with one side usually the one containing the more complicated expression I use basic identities and algebraic manipulations to arrive at the other side I CAUTION do not treat identities to be established as if they were equations cannot gtxlt values to each side etc sin 0 sinB tan29 secze 1 I it is sometimes helpful to write one side in terms of sines and cosines I add or subtract fractions where appropriate I establishing identities occurs through trial amp error and LOTS of practice Section 62 SUM AND DIFFERENCE FORMULAS cosa 3 cosacos 3 sina sin 3 cosa 3 cosacos 3 sina sin 3 sina 3 sinacos 3 cosa sin 3 sina 3 sinacos3 cosa sin3 tana tan tana 3 3 1 tana tan 3 tana tan tana 3 3 1 tanatan3 FORMULAS for 63 DOUBLEANGLE FORMULAS sin29 2sin6cos6 cos 26 cos2 6 sin2 6 2tan 6 tan26 2 1 tan 6 Variations cos26 1 2sin2 6 cos26 2cos2 6 1 Sinz 6 1 cos26 These are derived directly from the 2 sum formulas COSZ 6 1 cos26 Ex sin26 sin6 6 2 sin9cos9 cosOsinO sin9cos9 sin9cos9 tanz 6 1 C0829 2sin9cos6 1cos26 HALFANGLE FORMULAS Variations a 1 cosa Slnii 2 a 1 cosoc tan 39 smoc a 1 cosa cos i T tan smoc 1 cosoc a 1 cosa tan i 1 cosa These are derived from the last three variations of the doubleangle formulas Ex let 26 a then 9 and we have 1cosa cos2 The choice of i depends on the quadrant of a 1cosa gt COSEi 2 2 These two do not depend on finding the quadrant of PARTIAL NOTES for 71 Solving Right Triangles To m a triangle means to find the lengths of all three sides and the measures of all three angles gt a right angle 900 gt there are 1800 in a triangle gt so a 3 900 2 a amp 3 are complimentary recall that cofunctions of complimentary angles are equal gt 02 a2 192 gt recall soh cah toa Angle of elevation the acute angle formed by a horizontal line and an observer s line of sight to an object above the horizontal Angle of depression the acute angle formed by a horizontal line and an observer s line of sight to an object below the horizontal EXAMPLES for 71 1 A right triangle contains an angle of radians If one leg is 3 meters what is the length of the hypotenuse Hint two answers are possible cos1 sinl 3 3 a 30031 3s1nl b 8 8 c 8 8 71 39 at 3 3 cosg smg d 3cos3 3sin3 2 A right triangle has a 4inch hypotenuse If one angle is 400 find the length of each leg 3 A 30foot ladder leaning against a vertical wall just reaches a window sill If the ladder makes an angle of 470 with the ground how high is the window sill 4 The angle of elevation of a monument is 300 at the instant it casts a shadow 800 feet long Calculate the height of the monument 5 A ramp for wheel chair accessibility has an angle of elevation of 100 and a final height of 10 feet How long is the ramp 6 In the right triangle ABC if hypotenuse c 1 and b X then tanacot 3 2 1 2 1 2 x b f cl 1 f 1 x x x e none of these 7 A straight trail with a uniform inclination of 200 leads from a hotel whose elevation is 10000 feet to a mountain lake at an elevation of 11000 feet What is the length of the trail 8 A sighting is taken of a statue which is 300 feet tall If the angle of elevation to the top of the statue is 300 how far is the observer from the base of the statue 9 A lifeguard is seated on a high platform so that her eyes are 7 meters above sea level Suddenly she spots the dorsal fin of a great white shark at a 40 angle of depression What is the horizontal distance between the base of the platform and the shark

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