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

2. Quadratic 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. Quadratic Trig Equations Solution: 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.

4. Quadratic Trig Equations 1 1 st solution: or for Ans:

5. Quadratic Trig Equationse.g. 2 Solve the equation for the interval, giving answers to 1 d.p. or The original problem has become 2 simple trig equations, so we again 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. )

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

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

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

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

10. Quadratic Trig Equations 1 Principal Solution: Solving for. Ans:

11. Quadratic Trig EquationsExercises giving the answers as exact fractions of. 2.Solve the equation for 1.Solve the equation for. Solution: Ans: Solution: Ans:

12. Use of Trig Identities e.g. The formula we use is sometimes called the Pythagorean Identity and we will prove it now. 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.

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

14. Use of Trig Identities 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. We have shown that this formula holds for any angle in a right angled triangle. 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. Use of Trig Identities 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.

16. Use of Trig Identities 1 Tip: Factorising is easier if the squared term is positive. orPrincipal values:

17. Use of Trig Identities 1 orPrincipal values: Ans: We just look at the graph!

18. Use of Trig IdentitiesA 2 nd Trig Identity Consider the right angled triangle ABC. c a b A BC Also,So,

19. Use of Trig Identities 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.

20. Use of Trig Identities for Principal value: rads. Add to get 2 nd solution: Ans:

21. Use of Trig Identities 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 or

22. Use of Trig Identities 2.Solve the equation for giving the answers correct to 3 significant figures. 1.Solve the equation for Exercises

23. Use of Trig IdentitiesSolutions 1.Solve the equation for Solution:Substitute in or We’ll collect the terms on the r.h.s. so that the squared term is positive.

24. Use of Trig Identities 1 Ans: Principal values:  for or

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

26. Quadratic Trig Equations

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

31. Quadratic Trig Equations and Use of Identities Principal Solution: Solving for. Ans:

32. Quadratic Trig Equations and Use of Identities 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. Quadratic Trig Equations and Use of Identities 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. Quadratic Trig Equations and Use of Identities or Principal values: Ans: We just look at the graph!

35. Quadratic Trig Equations and Use of Identities 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. Quadratic Trig Equations and Use of Identities 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 or