1. “Teach A Level Maths” Vol. 2: A2 Core Modules
33: The equation
© Christine Crisp

2. Module C3 Module C4 Edexcel AQA OCR MEI/OCR
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3. Can you see why one of these equations is easy to solve and the other takes much more work ?
(b)
Both have 2 trig ratios but (a) can be solved by dividing by
We get
This is a simple equation and can now be solved.

4. Can you see why one of these equations is easy to solve and the other takes much more work ?
(b)
If we try the same method with (b), we get
This is no better than the original equation as we still have 2 trig ratios.

5. However, we saw in the previous section that
so the equation
can be written as
Dividing by 5:
This is of the form where so we can find solutions for and then find x by adding to each one.

6. e.g. 1 Solve the following equation giving the solutions in the interval correct to 1 d.p.
Let
Coef. of :
Coef. of :

7. Substituting into the l.h.s. of the equation:
Beware !
Don’t find x at this stage.
We have NOT
The 2nd solution will be wrong if we use the x value to try to find it.
At this stage we need to get all the solutions for .
So,
( Subtract from each part )

8. We sketch the usual cosine graph:
Outside the required interval

9. We sketch the usual cosine graph:
Add :
ANS: x is
( 1 d.p. )

10. SUMMARY
To solve the equation
Write the l.h.s. in one of the forms
Calculate the interval for using the one given for x, where or
Solve the equation to find making sure you find all the solutions.
Find the values of x.
N.B. for , add a and for
, subtract a.

11. Exercise
1(a) Write in the form where R and a are exact.
(b) Solve the equation
for
2. Solve the equation for
( Notice the different letter in the equation. You need to be able to cope with a switch of letters. )

12. Solutions:
1(a)
Coef. of :
Coef. of :
So,

13. Solutions:
for
1(b)
Solve
so, the equation becomes

14. Solutions:
2. Solve
Let
Coef. of :
Coef. of :
So,
So,
becomes

15. Solutions:
for
Solve

17. 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.

18. (a)
(b)
Both have 2 trig ratios but (a) can be solved by dividing by
We get
This is a simple equation and can now be solved.
Think about these 2 equations.

19. If we try the same method with (b), we get
This is no better than the original equation as we still have 2 trig ratios.
so the equation
can be written as
Dividing by 5:
However, we saw in the previous section that
This is now a simple equation which can be solved.

20. e.g. 1 Solve the following equation giving the solutions in the interval correct to 1 d.p.
Let
Coef. of :

21. Substituting into the l.h.s. of the equation:
At this stage we need to get all the solutions for .
So,
Beware !
Don’t find x at this stage.
We have NOT
The 2nd solution will be wrong if we use the x value to try to find it.
( Subtract from each part )

22. We sketch the usual cosine graph:
Add :
ANS: x is
( 1 d.p. )