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

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. Harder Trig Equations e.g.1 Solve the equation for the interval 1 st solution: Sketch to find the 2 nd solution: Solution: Let so, ( Once we have 2 adjacent solutions we can add or subtract to get the others. ) 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. Harder Trig Equations So, For, the other solutions are So, N.B. We must get all the solutions for x before we find. Alternate solutions for are NOT apart. for

5. Harder Trig Equations e.g. (a) for Use and We can use the same method for any function of. e.g. (b) for Use and e.g. (c) for

6. Harder Trig EquationsSUMMARY  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. Harder Trig Equations 1 Exercise So, 1.Solve the equation for Solution: Let Principal value:

8. Harder Trig Equations e.g. If an exact value is not required, then switch the calculator to radian mode and get (3 d.p.) We sometimes need to give answers in radians. If so, we may be asked for exact fractions of. Principal value is Tip: If you don’t remember the fractions of, use your calculator in degrees and then convert to radians using radians So, from the calculatorrads.

9. Harder Trig Equations e.g. 2 Solve the equation giving exact answers in the interval. The use of always indicates radians. Solution: Let ( or ) 1 st solution is For “tan” equations we usually keep adding to find more solutions, but working in radians we must remember to add. So,

10. Harder Trig Equations Solution: Let e.g. 3 Solve the equation for the interval. Principal value: rads. Sketch for a 2 nd value:

11. Harder Trig Equations 1 2 nd value: repeats every, so we add to the principal value to find the 3 rd solution: for Ans:

12. Harder Trig Equations e.g. 4 Solve the equation for giving the answers correct to 2 decimal places. We need to use radians but don’t need exact answers, so we switch the calculator to radian mode. Solution: We can’t let so we use a capital X ( or any another letter ). Let so Principal value: Sketch for the 1st solution that is in the interval: 2 1

13. Harder Trig Equations 1 1 st solution is 2 nd solution is Multiply by 2 : Ans: for ( 2 d.p.)

14. Harder Trig Equations 1.Solve the equation for giving the answers as exact fractions of. 2. Solve the equation for giving answers correct to 1 decimal place. Exercise

15. Harder Trig Equations 1.Solve the equation for Principal value: Solution: Let Add : Solutions

16. Harder Trig Equations 2. Solve the equation for giving answers correct to 1 decimal place. Principal value: Sketch for the 2 nd solution: Solutions  Solution: Let

17. Harder Trig Equations 1 The 2 nd value is too large, so we subtract for Ans:Add :

18. Harder Trig Equations

19. 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. 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. Harder Trig Equations e.g. 1 Solve the equation for the interval 1 st solution: Sketch to find the 2 nd solution: Solution: Let so, ( Once we have 2 adjacent solutions we can add or subtract to get the others. ) 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. Harder Trig Equations So, For, the other solutions are So, N.B. We must get all the solutions for x before we find. Alternate solutions for are NOT apart. for

23. Harder Trig Equations e.g. (a) for Use and We can use the same method for any function of. e.g. (b) for Use and e.g. (c) for

24. Harder Trig Equations The use of always indicates radians. e.g. 2 Solve the equation giving exact answers in the interval. Solution: Let ( or ) 1 st solution is For “tan” equations we usually keep adding to find more solutions, but working in radians we must remember to add.

25. Harder Trig Equations Solution: Let e.g. 3 Solve the equation for the interval. Principal value: rads. Sketch for a 2 nd solution:

26. Harder Trig Equations 2 nd value: repeats every, so we add to the 1 st value: for Ans: So,

27. Harder Trig Equations e.g. 4 Solve the equation for giving the answers correct to 2 decimal places. We need to use radians but don’t need exact answers, so we switch the calculator to radian mode. Solution: We can’t let so we use a capital X ( or any another letter ). Let so Principal value: Sketch for 1st solution that is in the interval: 2 1

28. Harder Trig Equations 1 st solution is 2 nd solution is Multiply by 2 : Ans: for ( 2 d.p.)