1. Further Trig identities Chapter 7

2. After completing this chapter you should be able to: Use the addition formulae Use the double angle formulae Write expressions of the form acosϴ ± bsinϴ in the form Rcos(ϴ ± α) and/or Rsin(ϴ ± α) Use the factor formulae Use all of the above to solve equations and prove identities

3. The Addition Formulae These are listed on page 107, once we’ve looked at them we’ll use them Oh goody that’s so exciting I hear you all thinking (apart from Gamal who insists on telling us all)

5. Use the double angle formulae to solve more equations and prove more identities Write down what you remember about tan2 ϴ and take it from there

6. Show sin3A ≡ 3sinA – 4sin ³ A Write down what happens if you use sin(2A + A)

7. examples 14 and 15 take your skills further exercise 7C page 118

8. Write expressions of the form acosϴ ± bsinϴ in the form Rcos(ϴ ± α) and/or Rsin(ϴ ± α)

9. to evaluate R and α you should expand the expression including R into it’s equivalent addition formulae e.g. Rsin(ϴ - α) = Rsinϴcosα - Rcosϴsinα Rsin(ϴ + α) = Rsinϴcosα + Rcosϴsinα Rcos(ϴ + α) = Rcosϴcosα - Rsinϴsinα Rcos(ϴ - α) = Rcosϴcosα + Rsinϴsinα and then equate corresponding coefficients to find R and α

10. example 17 a)express sinx - √3cosx in the form Rsin(x - α) b)plot the graph..... check the book out for this R² = a² + b² R = √(1² + ( √3) ²) = 2 the quick way gives us

12. express 2cosϴ + 5sinϴ in the form Rcos(ϴ - α) the check tells us R² = a² + b² R = √(2² + 5²) = √29 this gives us 2cos ϴ + 5sin ϴ = √29cos( ϴ - 68.2)

14. 2cosϴ + 5sinϴ = 3 √29cos(ϴ - 68.2) = 3 cos(ϴ - 68.2) = 3 ÷ √29 ϴ - 68.2 = cos -1 (3 ÷ √29) ϴ - 68.2 = -56.1° and ϴ - 68.2 = 56.1° ϴ = 12.1 °, 124.3 ° ( to the nearest 0.1 °) lets try the next part and solve the equation 2cosϴ + 5sinϴ = 3 for 0 < ϴ < 360°

15. The Factor Formulae these formulae are derived from the addition formulae

16. the examples show the standard manipulation of these formulae which we have looked at quite a bit now, so read them through and then try exercise 7D