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

2. Module C2
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3. 1st solution: e.g.1 Solve the equation for the interval
Solution: Let so,
We can already solve this equation BUT the interval for x is not the same as for .
There will be 4 solutions ( 2 for each cycle ).
1st solution:
( Once we have 2 adjacent solutions we can add or subtract to get the others. )
Sketch to find the 2nd solution:

4. for
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.

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

6. SUMMARY
Solving Harder Trig Equations
Replace the function of by x.
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. Exercise Solve the equation for Solution: Let Principal value: So, 1-1
So,

8. We sometimes need to give answers in radians
We sometimes need to give answers in radians. If so, we may be asked for exact fractions of .
e.g.
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 calculator
rads.
If an exact value is not required, then switch the calculator to radian mode and get (3 d.p.)

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

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

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

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

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

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

15. Solutions
Solve the equation for
Solution: Let
Principal value:
Add :

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

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

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. 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. 1st solution: e.g. 1 Solve the equation for the interval
Sketch to find the 2nd 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. So,
For , the other solutions are
N.B. We must get all the solutions for x before we find
. Alternate solutions for are NOT apart.
for

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

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

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

27. Solution: We can’t let so we use a capital X
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. 1st solution is
2nd solution is
Multiply by 2: Ans:
for
( 2 d.p.)