1. 43: Quadratic Trig Equations and Use of Identities
“Teach A Level Maths” Vol. 1: AS Core Modules
43: Quadratic Trig Equations and Use of Identities
© Christine Crisp

2. Module C2
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3. Quadratic equation so 2 solutions!
e.g. 1 Solve the equation for the interval
This is the shorthand notation for
Solution:
Quadratic equation so 2 solutions!
Square rooting: or
The original problem has become 2 simple trig equations, so we solve in the usual way.

4. or
for
1st solution:
1st solution: 1 -1
Ans:

5. e.g. 2 Solve the equation for the interval , giving answers to 1 d.p.
Solution: Let Then,
This is a quadratic equation, so it has 2 solutions.
( Method: Try to factorise; if there are no factors, use the formula or complete the square. )
Common factor: or
The original problem has become 2 simple trig equations, so we again solve in the usual way.

6. This is easy! We can just use the sketch.or
for
This is easy! We can just use the sketch. 1 -1
Principal value:

7. This is easy! We can just use the sketch.or
for
This is easy! We can just use the sketch. 1 -1
Principal value:

8. This is easy! We can just use the sketch.or
for
This is easy! We can just use the sketch. 1 -1
Principal value:
Ans:

9. e.g. 3 Solve the equation for the interval , giving exact answers.
Solution: Let Then,
Factorising: or 1 -1
The graph of
shows that always lies between -1 and +1 so,
has
no solutions for .

10. Solving for
Principal Solution: -1 1
Ans:

11. Exercises
Solve the equation for .
Solution:
Ans:
giving the answers as exact fractions of .
2. Solve the equation for
Solution:
Ans:

12. We can only solve a trig equation if we can reduce it to one, or more, of the following:
So, if we have an equation with and
e.g.
. . . we need a formula that will change one of these trig ratios into a function of the other.
The formula we use is sometimes called the Pythagorean Identity and we will prove it now.

13. c a b A B C Proof of the Pythagorean Identity.
Consider the right angled triangle ABC.
Using Pythagoras’ theorem:
Divide by :
But and

14. We have shown that this formula holds for any angle
in a right angled triangle.
However, because of the symmetries of and
, it actually holds for any value of .
A formula like this which is true for any value of the variable is called an identity.
Identity symbol
Identity symbols are normally only used when we want to stress that we have an identity. In the trig equations we use an = sign.

15. for giving answers correct to 1 d.p.
e.g.4 Solve the equation
for giving answers correct to 1 d.p.
Method: We use the identity to replace in the equation.
Solution:
Rearranging:
Substitute in
We always use the identity to substitute for the squared term.
Let and multiply out the brackets:

16. Tip: Factorising is easier if the squared term is positive.
Principal values: 1 -1

17. We just look at the graph!or
Principal values:
We just look at the graph! 1 -1
Ans:

18. c a b A B C A 2nd Trig Identity
Consider the right angled triangle ABC.
Also,
So,

19. We now have one simple trig equation.
e.g.5 Solve the equation for
giving exact answers.
Warning! We notice that there are 2 trig ratios but no squared term. We MUST NOT try to square root the Pythagorean identity since
DOES NOT GIVE
Method: Divide by
Since is not zero, we can divide by it.
We can now use the identity
We now have one simple trig equation.

20. for
rads.
Principal value:
Add to get 2nd solution:
Ans:

21. SUMMARY
With a quadratic equation, if there is only 1 trig ratio
Replace the ratio by c, s or t as appropriate.
Collect the terms with zero on one side of the equation.
Factorise the quadratic and solve the resulting 2 trig equations.
If there are 2 trig ratios, use
to substitute for or
if there are no squared terms. or

22. Exercises
Solve the equation for
2. Solve the equation for
giving the answers correct to 3 significant figures.

23. Solutions
Solve the equation for
Solution:
Substitute in
We’ll collect the terms on the r.h.s. so that the squared term is positive. or

24. for or
Principal values: Þ 1 -1
Ans:

25. Solutions
2. Solve the equation for
giving the answers correct to 3 significant figures.
Solution:
Divide by :
Substitute using
Principal value: rads.
Add :
Ans: ( 3 s.f.)

27. 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.

28. Quadratic equation so 2 solutions!
Square rooting:
e.g. 1 Solve the equation for the interval or
This is the shorthand notation for
Quadratic equation so 2 solutions!
The original problem has become 2 simple trig equations, so we solve in the usual way.

29. or
The original problem has become 2 simple trig equations, so we solve in the usual way.
Solution: Let Then,
This is a quadratic equation, so it has 2 solutions.
Common factor:
( Method: Try to factorise; if there are no factors, use the formula or complete the square. )
e.g. 2 Solve the equation for the interval , giving answers to 1 d.p.

30. e.g. 3 Solve the equation for the interval , giving exact answers.
Factorising:
The graph of
Solution: Let Then,
shows that always lies between -1 and +1 so,
has
no solutions for .

31. Solving for
Ans:
Principal Solution:

32. e.g.
This formula is sometimes called a Pythagorean Identity ( since its proof uses Pythagoras’ theorem ).
We can only solve a trig equation if we can reduce it to one, or more, of the following: or
So, if we have an equation with and
. . . we need a formula that will change one of these trig ratios into a function of the other.
A formula like this which is true for any value of the variable is called an identity.

33. Let and multiply out the brackets: Solution: Rearranging:
Substitute in
e.g.4 Solve the equation
for giving answers correct to 1 d.p.
We always use the identity to substitute for the squared term.
Method: We use the identity to replace in the equation.

34. or
Principal values:
Ans:
We just look at the graph!

35. We now have one simple trig equation.
Method: Divide by
e.g.5 Solve the equation for
giving exact answers.
Warning! We notice that there are 2 trig ratios but no squared term. We MUST NOT try to square root the Pythagorean identity since
DOES NOT GIVE
We can now use the identity
Since is not zero, we can divide by it.
We now have one simple trig equation.

36. SUMMARY
With a quadratic equation, if there is only 1 trig ratio
Replace the ratio by c, s or t as appropriate.
Collect the terms with zero on one side of the equation.
Factorise the quadratic and solve the resulting 2 trig equations.
If there are 2 trig ratios, use or
if there are no squared terms.
to substitute for