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Further Trigonometric Identities

Further Trigonometric Identities. and their Applications. Introduction. This chapter extends your knowledge of Trigonometrical identities You will see how to solve equations involving combinations of sin, cos and tan

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Further Trigonometric Identities

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  1. Further Trigonometric Identities and their Applications

  2. Introduction • This chapter extends your knowledge of Trigonometrical identities • You will see how to solve equations involving combinations of sin, cos and tan • You will learn to express combinations of these as a transformation of a single graph

  3. Teachings for Exercise 7A

  4. Further Trigonometric Identities and their Applications You need to know and be able to use the addition formulae Q By GCSE Trigonometry: 1 P 1 So the coordinates of P are: B A O M N So the coordinates of Q are: Q P 7A

  5. Further Trigonometric Identities and their Applications You need to know and be able to use the addition formulae Multiply out the brackets Rearrange ≡ 1 7A

  6. Further Trigonometric Identities and their Applications You need to know and be able to use the addition formulae Q You can also work out PQ using the triangle OPQ: Q 1 P 1 1 B P A B - A O M N 1 2bcCosA Sub in the values 2Cos(B - A) Group terms 2Cos(B - A) Cos (B – A) = Cos (A – B) eg) Cos(60) = Cos(-60) 2Cos(A - B) 7A

  7. Further Trigonometric Identities and their Applications You need to know and be able to use the addition formulae 2Cos(A - B) 2Cos(A - B) Subtract 2 from both sides 2Cos(A - B) Divide by -2 Cos(A - B) Cos(A - B) = CosACosB + SinASinB Cos(A + B) = CosACosB - SinASinB 7A

  8. Further Trigonometric Identities and their Applications You need to know and be able to use the addition formulae Cos(A - B) ≡ CosACosB + SinASinB Cos(A + B) ≡ CosACosB - SinASinB Sin(A + B) ≡ SinACosB + CosASinB Sin(A - B) ≡ SinACosB - CosASinB 7A

  9. Further Trigonometric Identities and their Applications Tan (A+B) You need to know and be able to use the addition formulae Show that: Rewrite Sin(A + B) = SinACosB + CosASinB Tan (A+B) Sin(A - B) = SinACosB - CosASinB Divide top and bottom by CosACosB Cos(A - B) = CosACosB + SinASinB Tan (A+B) Cos(A + B) = CosACosB - SinASinB Simplify each Fraction TanA + TanB Tan (A + B) Tan (A+B) 1 - TanATanB Tan θ 7A

  10. Further Trigonometric Identities and their Applications You need to know and be able to use the addition formulae Cos(A - B) ≡ CosACosB + SinASinB Cos(A + B) ≡ CosACosB - SinASinB Sin(A + B) ≡ SinACosB + CosASinB Sin(A - B) ≡ SinACosB - CosASinB Tan (A + B) You may be asked to prove either of the Tan identities using the Sin and Cos ones! Tan (A - B) 7A

  11. Further Trigonometric Identities and their Applications You need to know and be able to use the addition formulae Show, using the formula for Sin(A – B), that: Cos(A + B) ≡ CosACosB - SinASinB Sin(A - B) ≡ SinACosB - CosASinB A=45, B=30 Cos(A - B) ≡ CosACosB + SinASinB Sin(A + B) ≡ SinACosB + CosASinB Sin(45 - 30) ≡ Sin45Cos30 – Cos45Sin30 These can be written as surds Sin(A - B) ≡ SinACosB - CosASinB Sin(45 - 30) ≡ Tan (A + B) Multiply each pair Tan (A - B) Sin(45 - 30) ≡ Group the fractions up Sin(15) ≡ 7A

  12. Further Trigonometric Identities and their Applications You need to know and be able to use the addition formulae Given that: Find the value of: 5 3 < A < 270˚ A 4 Cos Use Pythagoras’ to find the missing side (ignore negatives) Tan is positive in the range 180˚ - 270˚ Tan(A+B) B 13 5 Tan (A + B) 12 y = Tanθ Use Pythagoras’ to find the missing side (ignore negatives) 90 180 270 360 Tan is negative in the range 90˚ - 180˚ 7A

  13. Further Trigonometric Identities and their Applications You need to know and be able to use the addition formulae Given that: Find the value of: Tan (A + B) Substitute in TanA and TanB < A < 270˚ Tan (A + B) Cos Work out the Numerator and Denominator Tan (A + B) Tan(A+B) Leave, Change and Flip Tan (A + B) Tan (A + B) Simplify Tan (A + B) Although you could just type the whole thing into your calculator, you still need to show the stages for the workings marks… 7A

  14. Further Trigonometric Identities and their Applications You need to know and be able to use the addition formulae Given that: Express Tanx in terms of Tany… Rewrite the sin and cos parts Multiply out the brackets Divide all by cosxcosy Simplify Subtract 3tanxtany Subtract 2tany Factorise the left side Divide by (2 – 3tany) 7A

  15. Teachings for Exercise 7B

  16. Further Trigonometric Identities and their Applications You can express sin2A, cos 2A and tan2A in terms of angle A, using the double angle formulae Sin(A + B) ≡ SinACosB + CosASinB Replace B with A Sin(A + A) ≡ SinACosA + CosASinA Simplify Sin2A ≡ 2SinACosA Sin2A ≡ SinACosA Sin4A ≡ 2Sin2ACos2A 2A  4A ÷ 2 Sin2A ≡ 2SinACosA x3 2A = 60 3Sin2A ≡ 6SinACosA Sin60 ≡ 2Sin30Cos30 7B

  17. Further Trigonometric Identities and their Applications You can express sin2A, cos 2A and tan2A in terms of angle A, using the double angle formulae Cos(A + B) ≡ CosACosB-SinASinB Replace B with A Cos(A + A) ≡ CosACosA-SinASinA Simplify Cos2A ≡ Co Cos2A ≡ Co Replace Cos2A with (1 – Sin2A) Replace Sin2A with (1 – Cos2A) Cos2A ≡ (1 Cos2A ≡ Co1 - Co Cos2A ≡ 1 Cos2A ≡ 2Co 7B

  18. Further Trigonometric Identities and their Applications You can express sin2A, cos 2A and tan2A in terms of angle A, using the double angle formulae Tan (A + B) Replace B with A Tan (A + A) Simplify Tan 2A Tan 2A Tan 60 ÷ 2 2A = 60 Tan 2A x 2 Tan A 2A = A 2Tan 2A 7B

  19. Further Trigonometric Identities and their Applications You can express sin2A, cos 2A and tan2A in terms of angle A, using the double angle formulae Rewrite the following as a single Trigonometric function: 2θ θ Replace the first part Rewrite 7B

  20. Further Trigonometric Identities and their Applications You can express sin2A, cos 2A and tan2A in terms of angle A, using the double angle formulae Show that: Can be written as: Double the angle parts Replace cos4θ The 1s cancel out 7B

  21. Further Trigonometric Identities and their Applications You can express sin2A, cos 2A and tan2A in terms of angle A, using the double angle formulae Given that: Find the exact value of: 4 x 3 Use Pythagoras’ to find the missing side (ignore negatives) Cosx is positive so in the range 270 - 360 Therefore, Sinx is negative y = Cosθ Sin2x ≡ 2SinxCosx 90 180 270 360 Sub in Sinx and Cosx Sin2x = 2 Work out and leave in surd form y = Sinθ Sin2x = 7B

  22. Further Trigonometric Identities and their Applications You can express sin2A, cos 2A and tan2A in terms of angle A, using the double angle formulae Given that: Find the exact value of: 4 x 3 Use Pythagoras’ to find the missing side (ignore negatives) Cosx is positive so in the range 270 - 360 Therefore, Tanx is negative Tan 2x y = Cosθ 90 180 270 360 Sub in Tanx Tan 2x Work out and leave in surd form y = Tanθ 90 180 270 360 7B

  23. Teachings for Exercise 7C

  24. Further Trigonometric Identities and their Applications The double angle formulae allow you to solve more equations and prove more identities Prove the identity: Divide each part by tanθ Rewrite each part 7C

  25. Further Trigonometric Identities and their Applications The double angle formulae allow you to solve more equations and prove more identities By expanding: Show that: Replace A and B Replace Sin2A and Cos 2A Multiply out Replace cos2A Multiply out Group like terms 7C

  26. Further Trigonometric Identities and their Applications The double angle formulae allow you to solve more equations and prove more identities Given that: Eliminate θ and express y in terms of x… Replace Cos2θ and Sinθ Multiply by 4 and Subtract 3 Divide by 3 Multiply by -1 Subtract 3, divide by 4 Multiply by -1 7C

  27. Further Trigonometric Identities and their Applications The double angle formulae allow you to solve more equations and prove more identities Solve the following equation in the range stated: (All trigonometrical parts must be in terms x, rather than 2x) Replace cos2x Multiply out the bracket Group terms Factorise or Solve both pairs y = Cosθ Remember to find additional answers! 90 180 270 360 7C

  28. Teachings for Exercise 7D

  29. Further Trigonometric Identities and their Applications You can write expressions of the form acosθ + bsinθ, where a and b are constants, as a sine or cosine function only Show that: Can be expressed in the form: So: Replace with the expression Compare each term – they must be equal! So in the triangle, the Hypotenuse is R… R = 5 Inverse Cos Find the smallest value in the acceptable range given 7D

  30. Further Trigonometric Identities and their Applications You can write expressions of the form acosθ + bsinθ, where a and b are constants, as a sine or cosine function only Show that you can express: In the form: So: Replace with the expression Compare each term – they must be equal! R = 2 Divide by 2 Inverse cos Find the smallest value in the acceptable range 7D

  31. Further Trigonometric Identities and their Applications You can write expressions of the form acosθ + bsinθ, where a and b are constants, as a sine or cosine function only Show that you can express: In the form: So: Sketch the graph of: = Sketch the graph of: 1 Start out with sinx π/2 π 3π/2 2π -1 Translate π/3 units right 1 π/3 π/2 π 4π/3 3π/2 2π -1 2 Vertical stretch, scale factor 2 1 π/2 π 2π -1 π/3 4π/3 3π/2 At the y-intercept, x = 0 -2 7D

  32. Further Trigonometric Identities and their Applications You can write expressions of the form acosθ + bsinθ, where a and b are constants, as a sine or cosine function only Express: in the form: So: Replace with the expression Compare each term – they must be equal! R = √29 Divide by √29 Inverse cos Find the smallest value in the acceptable range 7D

  33. Further Trigonometric Identities and their Applications You can write expressions of the form acosθ + bsinθ, where a and b are constants, as a sine or cosine function only Solve in the given range, the following equation: Divide by √29 Inverse Cos Remember to work out other values in the adjusted range We just showed that the original equation can be rewritten… Add 68.2 (and put in order!) Hence, we can solve this equation instead! y = Cosθ -56.1 56.1 303.9 Remember to adjust the range for (θ – 68.2) -90 90 180 270 360 7D

  34. Further Trigonometric Identities and their Applications Rcos(θ – α) chosen as it gives us the same form as the expression You can write expressions of the form acosθ + bsinθ, where a and b are constants, as a sine or cosine function only Find the maximum value of the following expression, and the smallest positive value of θ at which it arises: Replace with the expression Compare each term – they must be equal! R = 13 Divide by 13 Max value of cos(θ - 22.6) = 1 Inverse cos Overall maximum therefore = 13 Find the smallest value in the acceptable range Cos peaks at 0 θ = 22.6 gives us 0 7D

  35. Further Trigonometric Identities and their Applications You can write expressions of the form acosθ + bsinθ, where a and b are constants, as a sine or cosine function only Whichever ratio is at the start, change the expression into a function of that (This makes solving problems easier) Remember to get the + or – signs the correct way round! 7D

  36. Teachings for Exercise 7E

  37. Further Trigonometric Identities and their Applications You can express sums and differences of sines and cosines as products of sines and cosines by using the ‘factor formulae’ You get given all these in the formula booklet! 7E

  38. Further Trigonometric Identities and their Applications You can express sums and differences of sines and cosines as products of sines and cosines by using the ‘factor formulae’ Using the formulae for Sin(A + B) and Sin (A – B), derive the result that: 1) 2) Add both sides together (1 + 2) Let (A+B) = P Let (A-B) = Q 1) 1) 2) 2) 1 + 2 1 - 2 Divide by 2 Divide by 2 7E

  39. Further Trigonometric Identities and their Applications You can express sums and differences of sines and cosines as products of sines and cosines by using the ‘factor formulae’ Show that: P = 105 Q = 15 Work out the fraction parts Sub in values for Cos60 and Sin45 Work out the right hand side 7E

  40. Further Trigonometric Identities and their Applications You can express sums and differences of sines and cosines as products of sines and cosines by using the ‘factor formulae’ Solve in the range indicated: P = 4θ Q = 3θ Work out the fractions Set equal to 0 Either the cos or sin part must equal 0… Adjust the range Inverse cos Solve, remembering to take into account the different range Once you have all the values from 0-2π, add 2π to them to obtain equivalents… 0 y = Cosθ π/2 3π/2 2π Multiply by 2 and divide by 7 7E

  41. Further Trigonometric Identities and their Applications You can express sums and differences of sines and cosines as products of sines and cosines by using the ‘factor formulae’ Solve in the range indicated: P = 4θ Q = 3θ Work out the fractions Set equal to 0 Either the cos or sin part must equal 0… Adjust the range Inverse sin Solve, remembering to take into account the different range Once you have all the values from 0-2π, add 2π to them to obtain equivalents 0 y = Sinθ 3π/2 π/2 2π Multiply by 2 7E

  42. Further Trigonometric Identities and their Applications You can express sums and differences of sines and cosines as products of sines and cosines by using the ‘factor formulae’ Prove that: In the numerator: Ignore sin(x + y) for now… Use the identity for adding 2 sines P = x + 2y Q = x Simplify Fractions Bring back the sin(x + y) we ignored earlier Factorise Numerator: 7E

  43. Further Trigonometric Identities and their Applications You can express sums and differences of sines and cosines as products of sines and cosines by using the ‘factor formulae’ Prove that: In the denominator: Ignore cos(x + y) for now… Use the identity for adding 2 cosines P = x + 2y Q = x Simplify Fractions Bring back the cos(x + y) we ignored earlier Factorise Numerator: Denominator: 7E

  44. Further Trigonometric Identities and their Applications You can express sums and differences of sines and cosines as products of sines and cosines by using the ‘factor formulae’ Prove that: Replace the numerator and denominator Cancel out the (2cosy + 1) brackets Use one of the identities from C2 Numerator: Denominator: 7E

  45. Summary • We have extended the range of techniques we have for solving trigonometrical equations • We have seen how to combine functions involving sine and cosine into a single transformation of sine or cosine • We have learnt several new identities

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