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Trigonometry

Trigonometry. Exact Values. Angles greater than 90 o. Useful Notation & Area of a triangle. Using Area of Triangle Formula. Sine Rule Problems. Cosine Rule Problems. Mixed Problems. Starter Questions. Exact Values. Learning Intention. Success Criteria.

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Trigonometry

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  1. Trigonometry Exact Values Angles greater than 90o Useful Notation & Area of a triangle Using Area of Triangle Formula Sine Rule Problems Cosine Rule Problems Mixed Problems

  2. Starter Questions

  3. Exact Values Learning Intention Success Criteria • Recognise basic triangles and exact values for sin, cos and tan 30o, 45o, 60o . • To build on basic trigonometry values. • Calculate exact values for problems.

  4. 60º 2 2 2 60º 60º 60º 2 Exact Values Some special values of Sin, Cos and Tan are useful left as fractions, We call these exact values 30º 3 1 This triangle will provide exact values for sin, cos and tan 30º and 60º

  5. Exact Values 3 2 ½ 1 0 3 2 1 ½ 0 0 3

  6. Exact Values Consider the square with sides 1 unit 45º 2 1 1 45º 1 1 We are now in a position to calculate exact values for sin, cos and tan of 45o

  7. Exact Values 3 2 1 2 ½ 1 0 3 2 1 2 1 ½ 0 0 1 3

  8. Exact Values Now try Exercise 1 Ch8 (page 94)

  9. Starter Questions

  10. Angles Greater than 90o Learning Intention Success Criteria • Find values of sine, cosine and tangent over the range 0o to 360o. • Introduce definition of sine, cosine and tangent over 360o using triangles with the unity circle. • 2. Recognise the symmetry and equal values for sine, cosine and tangent.

  11. r y x P(x,y) y x Angles Greater than 90o We will now use a new definition to cater for ALL angles. New Definitions y-axis r Ao x-axis O

  12. Trigonometry Angles over 900 Example 1 The radius line is 2cm. The point (1.2, 1.6). Find sin cos and tan for the angle. (1.2, 1.6) Check answer with calculator 53o

  13. Trigonometry Angles over 900 Example 1 Check answer with calculator The radius line is 2cm. The point (-1.8, 0.8). Find sin cos and tan for the angle. (-1.8, 0.8) 127o

  14. What Goes In The Box ? Write down the equivalent values of the following in term of the first quadrant (between 0o and 90o): • Sin 300o • Cos 360o • Tan 330o • Sin 380o • Cos 460o • Sin 135o • Cos 150o • Tan 135o • Sin 225o • Cos 270o - sin 60o sin 45o cos 0o -cos 45o - tan 30o -tan 45o sin 20o -sin 45o - cos 80o -cos 90o

  15. Trigonometry Angles over 900 Now try Exercise 2 Ch8 (page 97)

  16. Trigonometry Angles over 900 Extension for unit 2 Trigonometry GSM Software

  17. Angles Greater than 90o Two diagrams display same data in a different format Sin +ve (0,1) All +ve 180o - xo (1,0) (-1,0) 360o - xo 180o + xo (0,-1) Cos +ve Tan +ve

  18. Starter Questions 3cm 8cm

  19. Area of a Triangle Learning Intention Success Criteria • Be able to label a triangle properly. • To show the standard way of labelling a triangle. • 2. Find the area of a triangle using basic trigonometry knowledge. • 2. Find the area of a triangle using basic trigonometry knowledge.

  20. In Mathematics we have a convention for labelling triangles. Labelling Triangles B B a c C C b A A Small letters a, b, c refer to distances Capital letters A, B, C refer to angles

  21. Have a go at labelling the following triangle. Labelling Triangles E E d f F F e D D

  22. Example 1 : Find the area of the triangle ABC. Area of a Triangle B (i) Draw in a line from B to AC (ii) Calculate height BD 10cm 7.66cm 50o D (iii) Area C A 12cm

  23. Example 2 : Find the area of the triangle PQR. Area of a Triangle P (i) Draw in a line from P to QR (ii) Calculate height PS 12cm 7.71cm 40o S (iii) Area R Q 20cm

  24. Constructing Pie Charts Now try Exercise 3 Ch8 (page 99)

  25. Starter Questions

  26. Area of ANY Triangle Learning Intention Success Criteria • Know the formula for the area of any triangle. • 1. To explain how to use the Area formula for ANY triangle. • 2. Use formula to find area of any triangle given two length and angle in between.

  27. Co a b h Ao Bo c General Formula forArea of ANY Triangle Consider the triangle below: Area = ½ x base x height What does the sine of Ao equal Change the subject to h. h = b sinAo Substitute into the area formula

  28. Key feature To find the area you need to knowing 2 sides and the angle in between (SAS) The area of ANY triangle can be found by the following formula. Area of ANY Triangle B B a Another version c C C Another version b A A

  29. Example : Find the area of the triangle. Area of ANY Triangle The version we use is B B 20cm c C C 30o 25cm A A

  30. Example : Find the area of the triangle. Area of ANY Triangle The version we use is E 10cm 60o 8cm F D

  31. (1) 12.6cm 23o 15cm (2) 5.7m 71o 6.2m Key feature Remember (SAS) What Goes In The Box ? Calculate the areas of the triangles below: A =36.9cm2 A =16.7m2

  32. Area of ANY Triangle Now try Exercise 4 Ch8 (page 100)

  33. Starter Questions

  34. Sine Rule Learning Intention Success Criteria • Know how to use the sine rule to solve REAL LIFE problems involving lengths. • 1. To show how to use the sine rule to solve REAL LIFE problems involving finding the length of a side of a triangle .

  35. Works for any Triangle Sine Rule The Sine Rule can be used with ANY triangle as long as we have been given enough information. B a c C b A

  36. Consider a general triangle ABC. C b a P A B c The Sine Rule Deriving the rule Draw CP perpendicular to BA This can be extended to or equivalently

  37. a 10m 34o 41o Calculating Sides Using The Sine Rule Example 1 : Find the length of a in this triangle. B C A Match up corresponding sides and angles: Now cross multiply. Solve for a.

  38. 10m 133o 37o d Calculating Sides Using The Sine Rule Example 2 : Find the length of d in this triangle. D E C Match up corresponding sides and angles: Now cross multiply. Solve for d. = 12.14m

  39. 12cm b (1) (2) 47o 32o a 72o 16mm 93o What goes in the Box ? Find the unknown side in each of the triangles below: a = 6.7cm b = 21.8mm

  40. Sine Rule Now try Ex 6&7 Ch8 (page 103)

  41. Starter Questions

  42. Sine Rule Learning Intention Success Criteria • Know how to use the sine rule to solve problems involving angles. • 1. To show how to use the sine rule to solve problems involving finding an angle of a triangle .

  43. 45m 38m 23o Ao Calculating Angles Using The Sine Rule Example 1 : Find the angle Ao Match up corresponding sides and angles: Now cross multiply: Solve for sin Ao Use sin-1 0.463 to find Ao = 0.463

  44. 75m Bo 38m 143o Calculating Angles Using The Sine Rule Example 2 : Find the angle Bo Match up corresponding sides and angles: Now cross multiply: Solve for sin Bo Use sin-1 0.305 to find Bo = 0.305

  45. (1) 8.9m 100o (2) Ao 12.9cm Bo 14.5m 14o 14.7cm What Goes In The Box ? Calculate the unknown angle in the following: Ao = 37.2o Bo = 16o

  46. Sine Rule Now try Ex 8 & 9 Ch8 (page 106)

  47. Starter Questions

  48. Cosine Rule Learning Intention Success Criteria • Know when to use the cosine rule to solve problems. • 1. To show when to use the cosine rule to solve problems involving finding the length of a side of a triangle . • 2. Solve problems that involve finding the length of a side.

  49. Works for any Triangle Cosine Rule The Cosine Rule can be used with ANY triangle as long as we have been given enough information. B a c C b A

  50. The Cosine Rule The Cosine Rule generalises Pythagoras’ Theorem and takes care of the 3 possible cases for Angle A. A Consider a general triangle ABC. We require a in terms of b, c and A. B a2 = b2 + c2 a c A P C A x b - x a2 > b2 + c2 b A When A = 90o, CosA = 0 and reduces to a2 = b2 + c2 1 1 b When A > 90o, CosA is positive,  a2 > b2 + c2 2 2 a2 < b2 + c2 When A < 90o, CosA is negative,  a2 > b2 + c2 3 3 Deriving the rule • BP2 = a2 – (b – x)2 • Also: BP2 = c2 – x2 • a2 – (b – x)2 = c2 – x2 • a2 – (b2 – 2bx + x2) = c2 – x2 • a2 – b2 + 2bx – x2 = c2 – x2 • a2 = b2 + c2 – 2bx* • a2 = b2 + c2 – 2bcCosA Draw BP perpendicular to AC *Since Cos A = x/c  x = cCosA Pythagoras Pythagoras + a bit Pythagoras - a bit

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