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42: Harder Trig 42: Harder Trig Equations Equations © Christine Crisp Teach A Level Maths” Teach A Level Maths” Vol. 1: AS Core Vol. 1: AS Core Modules Modules
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Harder trig equations

Aug 12, 2015

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  1. 1. 42: Harder Trig Equations42: Harder Trig Equations Christine Crisp Teach A Level MathsTeach A Level Maths Vol. 1: AS Core ModulesVol. 1: AS Core Modules
  2. 2. Harder Trig Equations "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages" Module C2
  3. 3. Harder Trig Equations 360360 e.g.1 Solve the equation for the interval 180180 502sin = x 30=x50sin =x 1st solution: 180180 Sketch to find the 2nd solution: Solution: Let so,2=x 50sin =x ( Once we have 2 adjacent solutions we can add or subtract to get the others. ) 360 There will be 4 solutions ( 2 for each cycle ). We can already solve this equation BUT the interval for x is not the same as for .
  4. 4. Harder Trig Equations 0 180 360 xy sin= 50=y 30 150 150,302 == x So, 33036030210360150 == and 360360 xFor , the other solutions are So, 150,30,210,3302 == x 75,15,105,165 = N.B. We must get all the solutions for x before we find . Alternate solutions for are NOT apart. 360 50sin =x 360360 xfor
  5. 5. Harder Trig Equations e.g. (a) forc=4tan 1800 4=x 7200 xUse and We can use the same method for any function of . c= 2 cos e.g. (b) for 360360 180180 x 2 =xUse and 33030 x 30= xUse and c= )30sin( e.g. (c) for 3600
  6. 6. Harder Trig Equations SUMMARY Replace the function of by x. Solving Harder Trig Equations Write down the interval for solutions for x. Find all the solutions for x in the required interval. Convert the answers to values of .
  7. 7. Harder Trig Equations 0 180 360 xy cos= 1 -1 50=y Exercise 300 60 3600 7200 x 60=x50cos =x 360300,36060,300,602 ++== xSo, 330,210,150,30= 1. Solve the equation for502cos = 3600 Solution: Let 2=x 50cos =x Principal value: 660,420,300,602 == x
  8. 8. Harder Trig Equations e.g. 2 1 cos =x If an exact value is not required, then switch the calculator to radian mode and get (3 d.p.) c 7850=x We sometimes need to give answers in radians. If so, we may be asked for exact fractions of . Principal value is 4 Tip: If you dont remember the fractions of , use your calculator in degrees and then convert to radians using radians 180= So, from the calculator 45 2 1 cos == xx 4 = x rads.
  9. 9. Harder Trig Equations e.g. 2 Solve the equation giving exact answers in the interval . 0 13tan = The use of always indicates radians. Solution: Let 3=x 0 30 x 4 =x ( or ) 4 45 == x 12 9 , 12 5 , 12 = 3 4 1st solution is1tan =x For tan equations we usually keep adding to find more solutions, but working in radians we must remember to add . 180 4 9 , 4 5 , 4 3 == x 4 2, 4 , 4 3 ++==xSo,
  10. 10. Harder Trig Equations Solution: Let 4 +=x 2 1 cos =x e.g. 3 Solve the equation for the interval . 2 1 4 cos = + 20 == 45xPrincipal value: 2 1 cos =x 4 rads. 20 4 2 4 + x 44 9 x Sketch for a 2nd value:
  11. 11. Harder Trig Equations 0 2 xy cos= 1 -1 70 2 1 =y 4 2nd value: 4 7 4 2 =x 4 7 =x repeats every , so we add to the principal value to find the 3rd solution: 2 2xcos 44 9 x 2 1 cos =x for 2, 2 ,0 3 =Ans: 4 9 2 4 =+= x 4 9 , 4 7 , 44 =+= x 4 8 , 4 6 ,0 = += 4 x 2 2 3 1
  12. 12. Harder Trig Equations e.g. 4 Solve the equation for giving the answers correct to 2 decimal places. 40 2 sin = x 40 x We need to use radians but dont need exact answers, so we switch the calculator to radian mode. Solution: We cant let so we use a capital X ( or any another letter ). 2 x x = Let so 2 x X = 40sin =X 40 x 2 4 0 X Principal value: )410( c =X Sketch for the 1st solution that is in the interval: 2 1
  13. 13. Harder Trig Equations y 1 -1 40=y X Xy sin= 4120 5533 2 1st solution is c 5533= 2nd solution is c 4120 2 +== x X c 41202 2 == x X c 8725= 8725 Multiply by 2: Ans: 20 X40sin =X for cc 7411,117 =x ( 2 d.p.) = 2 x X
  14. 14. Harder Trig Equations 1. Solve the equation for12tan = 20 giving the answers as exact fractions of . 2. Solve the equation for250)60(sin = 180180 giving answers correct to 1 decimal place. Exercise
  15. 15. Harder Trig Equations 20 40 x 4 =x1tan =x 4 13 , 4 9 , 4 5 , 4 2 ==x 8 13 , 8 9 , 8 5 , 8 = 1. Solve the equation for12tan = 20 Principal value: Solution: Let 2=x 1tan =x Add : Solutions
  16. 16. Harder Trig Equations 180180 514=x 250sin =x 2. Solve the equation for250)60(sin = 180180 giving answers correct to 1 decimal place. 120240 x Principal value: 250sin =x Sketch for the 2nd solution: Solutions Solution: Let 60= x
  17. 17. Harder Trig Equations xy sin= y 1 -1 x 360 180 )5165514180(,51460 === x 514 )5165( 250=y The 2nd value is too large, so we subtract 360 250sin =x 120240 xfor 5194360516560 === x 574,5134 =Ans: 60Add :
  18. 18. Harder Trig Equations
  19. 19. Harder Trig Equations The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as Handouts with up to 6 slides per sheet.
  20. 20. Harder Trig Equations SUMMARY Replace the function of by x. Solving Harder Trig Equations Write down the interval for solutions for x. Find all the solutions for x in the required interval. Convert the answers to values of .
  21. 21. Harder Trig Equations 360360 e.g. 1 Solve the equation for the interval 180180 502sin = x 30=x50sin =x 1st solution: 180180 Sketch to find the 2nd solution: Solution: Let so,2=x 50sin =x ( Once we have 2 adjacent solutions we can add or subtract to get the others. ) 360 There will be 4 solutions ( 2 for each cycle ). We can already solve this equation BUT the interval for x is not the same as for .
  22. 22. Harder Trig Equations 150,302 == x So, 33036030210360150 == and 360360 xFor , the other solutions are So, 150,30,210,3302 == x 75,15,105,165 = N.B. We must get all the solutions for x before we find . Alternate solutions for are NOT apart. 360 50sin =x 360360 xfor xy sin= 50=y 150 30
  23. 23. Harder Trig Equations e.g. (a) forc=4tan 1800 4=x 7200 xUse and We can use the same method for any function of . c= 2 cos e.g. (b) for 360360 180180 x 2 =xUse and 33030 x 30= xUse and c= )30sin( e.g. (c) for 3600
  24. 24. Harder Trig Equations The use of always indicates radians. e.g. 2 Solve the equation giving exact answers in the interval . 0 13tan = Solution: Let 3=x 0 30 x 4 =x ( or ) 4 45 == x 4 9 , 4 5 , 4 3 ==x 12 9 , 12 5 , 12 = 3 4 1st solution is1tan =x For tan equations we usually keep adding to find more solutions, but working in radians we must remember to add . 180
  25. 25. Harder Trig Equations Solution: Let 4 +=x 2 1 cos =x e.g. 3 Solve the equation for the interval . 2 1 4 cos = + 20 == 45xPrincipal value: 2 1 cos =x 4 rads. 20 4 2 4 + x 44 9 x Sketch for a 2nd solution:
  26. 26. Harder Trig Equations 70 2 1 =y 4 2nd value: 4 7 4 7 =x 4 2 =x repeats every , so we add to the 1st value:2 2xcos 44 9 x 2 1 cos =x for 2, 2 ,0 3 =Ans: 4 9 2 4 =+= x 4 9 , 4 7 , 44 =+=x 4 8 , 4 6 ,0 = += 4 x 2 2 3 xy cos= So,
  27. 27. Harder Trig Equations e.g. 4 Solve the equation for giving the answers correct to 2 decimal places. 40 2 sin = x 40 x We need to use radians but dont need exact answers, so we switch the calculator to radian mode. Solution: We cant let so we use a capital X ( or any another letter ). 2 x x = Let so 2 x X = 40sin =X 40 x 2 4 0 X Principal value: )410( c =X Sketch for 1st solution that is in the interval: 2 1
  28. 28. Harder Trig Equations 40=y X Xy sin= 4120 5533 1st solution is c 5533= 2nd solution is c 4120 2 +== x X c 41202 2 == x X c 8725= 8725 Multiply by 2: Ans: 20 X40sin =X for cc 7411,117 =x ( 2 d.p.)