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

2. Module C2
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3. Solution: The calculator gives us the solution x =
e.g.1 Solve the equation
Solution: The calculator gives us the solution x =
BUT, by considering the graphs of and , we can see that there are many more solutions:
principal solution
Every point of intersection of and gives a solution ! In the interval shown there are 10 solutions, but in total there are an infinite number.
The calculator value is called the principal solution

4. Tip: Check that the solution from the calculator looks reasonable.
We will adapt the question to:
Solve the equation for
Solution: The first answer still comes from the calculator:
Sketch between 1 -1
Add the line
There are 2 solutions.
It’s important to show the scale.
Tip: Check that the solution from the calculator looks reasonable.
The symmetry of the graph . . .
. . . shows the 2nd solution is

5. e.g. 2 Solve the equation in the interval
Solution: The first answer from the calculator is
Sketch between 1 -1
Add the line
There are 2 solutions.
The symmetry of the graph . . .
shows the 2nd solution is

6. SUMMARY
where c is a constant
To solve or
for
Find the principal solution from a calculator.
Sketch one complete cycle of the trig function. For example sketch from to . or
Draw the line y = c.
Find the 2nd solution using symmetry

7. Solution: (a) ( from calculator )
Exercises
Solve the equations
(a) and (b) for
Solution: (a) ( from calculator ) 1 -1
The 2nd solution is

8. Solution: ( from calculator )
Exercises
(b)
Solution: ( from calculator ) 1 -1
The 2nd solution is

9. e.g. 3 Solve the equation in the interval
More Examples
e.g. 3 Solve the equation in the interval
giving answers correct to the nearest whole degree.
Solution: ( from the calculator ) 2 -2
The 2nd solution is

10. 2 -2
Notice that the period of is and there is only one solution to the equation in each interval of .
So all solutions to the equation can be found by repeatedly adding or subtracting to the first value.

11. So, to solve for
Principal solution:
First subtract
Now add to
and keep adding . . .
Ans:
This process is easy to remember, so to solve
there is no need to draw a graph.

12. Exercise
Solve the equation for
Solution: Principal value
Adding
Ans:

13. More Examples
e.g. 4 Solve the equation for
giving the answers correct to 2 d. p.
Solution:
implies radians
Switching the calculator to radians, we get
( Because of the interval, it’s convenient to sketch from to . )
2nd solution:
Ans:

14. Tip: Bracket a value if it is outside the interval.
More Examples
e.g. 5 Solve the equation for
Solution: ( from the calculator )
. . .
This value is outside the required interval
but we still use it to solve the equation.
Tip: Bracket a value if it is outside the interval.
We extend the graph to the left to show

15. e.g. 5 Solve the equation for
More Examples
e.g. 5 Solve the equation for
Solution: 1 -1
Since the period of the graph is this solution . . . is

16. e.g. 5 Solve the equation for
More Examples
e.g. 5 Solve the equation for
Solution: 1 -1
Symmetry gives the 2nd value for
The values in the interval are and

17. ) 360 ( £ x The graphs of and repeat every .
So, if more solutions are required we add ( or subtract ) to those we already have.
and
e.g. In the previous example, we had )
360 ( o x
For solutions in the interval ,
we also have

18. Solution: Principal value Method 1
e.g. 6 Solve for
Solution: Principal value
Method 1 1 -1
By symmetry,
Subtract from :
( is outside the interval )
Ans:

19. Solution: Principal value Method 2
e.g. 6 Solve for
Solution: Principal value
Method 2
The solution can be found by using the symmetry of about the y-axis 1 -1
Add to :
Ans:

20. SUMMARY
To solve or
Find the principal value from the calculator.
Sketch the graph of the trig function showing at least one complete cycle and including the principal value.
Find a 2nd solution using the graph.
Once 2 adjacent solutions have been found, add or subtract to find any others in the required interval.
To solve
Find the principal value from the calculator.
Add or subtract to find other solutions.

21. Exercises
1. Solve the equations ( giving answers correct to the nearest whole degree )
(a) for
(b) for

22. Solution: Principal value
Exercises
(a) for
Solution: Principal value 1 -1
By symmetry,
Ans:

23. Solution: Principal value
Exercises
(b) for
Solution: Principal value
Either:
Or: 1 -1 -1 1
Ans:

25. 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.

26. Solution: The first answer comes from the calculator:
Add the line
Solve the equation for
Sketch between
There are 2 solutions.
The symmetry of the graph . . .
. . . shows the 2nd solution is
e.g. 1

27. e.g. 2 Solve the equation in the interval
Solution: The first answer from the calculator is
Sketch between
Add the line
There are 2 solutions.
The symmetry of the graph . . .
shows the 2nd solution is

28. e.g. 3 Solve for
Principal solution:
This process is easy to remember, so to solve
there is no need to draw a graph.
First subtract
Now add to
and keep adding . . .
Ans:

29. e.g. 4 Solve the equation for
giving the answers correct to 2 d. p.
( Because of the interval, it’s convenient to sketch from to . )
Switching the calculator to radians, we get
Solution:
radians
2nd solution:
Ans:

30. Tip: Bracket a value if it is outside the interval.
Solution: ( from the calculator )
e.g. 5 Solve the equation for
This value is outside the required interval . . .
but we still use it to solve the equation.
Tip: Bracket a value if it is outside the interval.
We extend the graph to the left to show

31. Since the period of the graph is , the 1st solution in is
Symmetry gives the 2nd value as
Ans: ,

32. Solution: Principal value
e.g. 6 Solve for
By symmetry,
Method 1
Ans:
Subtract from :
( is outside the interval )

33. The solution can be found by using the symmetry of about the y-axis
Method 2
Ans:
Add to :

34. SUMMARY
To solve or
Once 2 adjacent solutions have been found, add or subtract to find any others in the required interval.
Find the principal value from the calculator.
Sketch the graph of the trig function showing at least one complete cycle and including the principal value.
Find a 2nd solution using the graph.
To solve
Add or subtract to find other solutions.