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Tso math trig graphs

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  • www.mathsrevision.com 03/23/13 www.mathsrevision.com We start by find the equation of a circle centre the origin. First draw set axises x,y and then label the origin O. Next we plot a point P say, which as coordinates x,y. Next draw a line from the origin O to the point P and label length of this line r. If we now rotate the point P through 360 degrees keep the Origin fixed we trace out a circle with radius r and centre O. Remembering Pythagoras’s Theorem from Standard grade a square plus b squared equal c squares we can now write down the equal of any circle with centre the origin.

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  • 1. Trigonometry GraphsS4 Credit Exact values for Sin Cos and Tan www.mathsrevision.com Angles greater than 90o Graphs of the form y = a sin xo Graphs of the form y = a sin bxo Graphs of the form y = a sin bxo + c Solving Trig Equations Special trig relationships created by Mr. Lafferty
  • 2. Starter QuestionsS4 Credit 1. Factorise x2 - 36 www.mathsrevision.com 2. A car depreciates at 20% each year. How much is it worth after 1 year if it cost £15 000 initially. 3. What is sin30o as a fraction. 23 Mar 2013 Created by Mr Lafferty Maths Dept
  • 3. Exact ValuesS4 Credit Learning Intention Success Criteria www.mathsrevision.com 1. To build on basic 1. Recognise basic triangles and trigonometry values. exact values for sin, cos and tan 30o, 45o, 60o . 2. Calculate exact values for problems. 23 Mar 2013 Created by Mr Lafferty Maths Dept
  • 4. Exact ValuesS4 Credit Some special values of Sin, Cos and Tan are useful left as fractions, We call these exact values www.mathsrevision.com 60º 30º 2 2 2 √3 60º 60º 60º 2 1 This triangle will provide exact values for sin, cos and tan 30º and 60º
  • 5. Exact ValuesS4 Credit x 0º 30º 45º 60º 90º www.mathsrevision.com √3 Sin xº 0 ½ 2 1 √3 Cos xº 1 2 ½ 0 1 Tan xº 0 3 √3 ∞
  • 6. Exact ValuesS4 Credit Consider the square with sides 1 unit www.mathsrevision.com 45º 1 1 √2 45º 1 1 We are now in a position to calculate exact values for sin, cos and tan of 45o
  • 7. Exact ValuesS5 Int2 x 0º 30º 45º 60º 90º www.mathsrevision.com 1 √3 Sin xº 0 ½ √2 2 1 √3 1 Cos xº 1 2 √2 ½ 0 1 Tan xº 0 3 1 √3 ∞
  • 8. Exact ValuesS5 Int2 www.mathsrevision.com Now try Ex 2.1 Ch11 (page 220) 23 Mar 2013 Created by Mr Lafferty Maths Dept
  • 9. Starter QuestionsS4 credit 1. True or false 2 + 3 × 7 = 35 www.mathsrevision.com 2. A house increases by 3% each year. How much is it worth after 1 years if it cost £40 000 initially. 3. What is the exact value of sin 45o. 23 Mar 2013 Created by Mr Lafferty Maths Dept
  • 10. Angles Greater than 90oS4 credit Learning Intention Success Criteria www.mathsrevision.com 1. Introduce definition of 1. Find values of sine, cosine sine, cosine and tangent and tangent over the range 0o over 360o using triangles to 360o. with the unity circle. 2. Recognise the symmetry and equal values for sine, cosine and tangent. 23 Mar 2013 Created by Mr. Lafferty Maths Dept.
  • 11. r y Angles Greater than 90o xS4 credit We will now use a new definition to cater for ALL angles. www.mathsrevision.com New Definitions y-axis y sin A = y P(x,y) r r x A cos A = r O o x x-axis y tan A = x Mar 23, 2013 www.mathsrevision.com 11
  • 12. TrigonometryS4 credit Example Angles over 900 The radius line is 2cm. (1.2, 1.6) The point (1.2, 1.6). www.mathsrevision.com Find sin cos and tan for the angle. 1.6 Check answer sin 53o = = 0.80 2 with calculator 1.2 cos 53 = o = 0.60 2 tan 53 = o 1.6 = 1.33 53o 23 Mar 2013 1.2 Created by Mr Lafferty Maths Dept
  • 13. TrigonometryS4 credit Example 1 Angles over 900 The radius line is 2cm. Check answer The point (-1.8, 0.8). with calculator www.mathsrevision.com Find sin cos and tan for the angle. (-1.8, 0.8) 0.8 sin127o = = 0.40 2 127o −1.8 cos127 =o = −0.90 2 0.8 tan127 = o = −0.44 23 Mar 2013 −1.8 Created by Mr Lafferty Maths Dept
  • 14. Trigonometry Summary of resultsS4 credit Example All Quadrants Calculate the ration for sin cos and tan for the angle values below. 90o www.mathsrevision.com 30o 210o 45o 225o Sin +ve All +ve 60o 240o 180o - xo xo 120o 300o 0o 180o 135o 315o 180o + xo 360o - xo 150o 330o Tan +ve Cos +ve Sin x Cos x Tan x 23 Mar 2013 Created by Mr Lafferty Maths Dept 270o
  • 15. What Goes In The Box ?S4 credit Write down the equivalent values of the following in term of the first quadrant (between 0o and 90o): www.mathsrevision.com 1) Sin 135o sin 45 o 1) Sin 300o - sin 60o 2) Cos 150o -cos 45o 2) Cos 360o cos 0o 3) Tan 135 o -tan 45o 3) Tan 330 o - tan 30o 4) Sin 225 o -sin 45o 4) Sin 380 o sin 20o -cos 90o - cos 80o 5) Cos 270o 5) Cos 460o
  • 16. TrigonometryS4 credit Angles over 900 www.mathsrevision.com Now try MIA Ch11 Ex3.1 Ch11 (page 222) 23 Mar 2013 Created by Mr Lafferty Maths Dept
  • 17. StarterS4 Credit 1. True or false www.mathsrevision.com 2x +7x+6 = ( 2x + 3)(x + 2) 2 2. A TV is reduced by 20% to £200. What was the original price. Q3. Solve (2x-1)(x-1) = 0 created by Mr. Lafferty
  • 18. Sine GraphS4 Credit Learning Intention Success Criteria www.mathsrevision.com 1. To investigate graphs of 1. Identify the key points the form for various graphs. y = a sin xo y = a cos xo y = tan xo created by Mr. Lafferty
  • 19. Key Features Sine Graph value at x = 90 Max Zeros at 0, 180o and 360o oS4 Credit Minimum value at x = 270o www.mathsrevision.com Key Features Domain is 0 to 360o (repeats itself every 360o) Maximum value of 1 Minimum value of -1 created by Mr. Lafferty
  • 20. What effect y = sinxo does the number at the front Sine Graph y = 2sinxo have on theS4 Credit graphs ? y = 3sinxo 3 y = 0.5sinxo y = -sinxo www.mathsrevision.com 2 1 0 90o 180o 270o 360o -1 What effect -2 does the negative sign have on the -3 graphs ? by Mr. Lafferty created
  • 21. Sine GraphS4 Credit y = a sin (x) www.mathsrevision.com For a > 1 stretches graph in the y-axis direction For a < 1 compresses graph in the y - axis direction For a - negative flips graph in the x – axis. created by Mr. Lafferty
  • 22. y = 5sinxo Sine Graph y = 4sinx o y = sinxoS4 Credit 6 y = -6sinxo www.mathsrevision.com 4 2 0 90o 180o 270o 360o -2 -4 -6 created by Mr. Lafferty
  • 23. Cosine Graphsat 90 and 270 Key Features Zeros o o Max value at x = 0o and 360oS4 Credit Minimum value at x = 180o www.mathsrevision.com Key Features Domain is 0 to 360o (repeats itself every 360o) Maximum value of 1 Minimum value of -1 created by Mr. Lafferty
  • 24. What effect y = cosxo does the number at the front Cosine 2cosx y= o have on theS4 Credit graphs ? y = 3cosxo 3 y = 0.5cosxo y = -cosxo www.mathsrevision.com 2 1 0 90o 180o 270o 360o -1 -2 -3 created by Mr. Lafferty
  • 25. y = 2cosxo Cosine Graph y = 4cosx oS4 Credit y = 6cosxo 6 y = 0.5cosxo y = -cosxo www.mathsrevision.com 4 2 0 90o 180o 270o 360o -2 -4 -6 created by Mr. Lafferty
  • 26. Key Features Tangent Graphs Zeros at 0 and 180oS4 Credit www.mathsrevision.com Key Features Domain is 0 to 180o (repeats itself every 180o) created by Mr. Lafferty
  • 27. Tangent GraphsS4 Credit www.mathsrevision.com created by Mr. Lafferty
  • 28. Tangent GraphS4 Credit y = a tan (x) www.mathsrevision.com For a > 1 stretches graph in the y-axis direction For a < 1 compresses graph in the y - axis direction For a - negative flips graph in the x – axis. created by Mr. Lafferty
  • 29. Period of a FunctionS4 Credit When a pattern repeats itself over and over, it is said to be periodic. www.mathsrevision.com Sine function has a period of 360o Let’s investigate the function y = sin bx created by Mr. Lafferty
  • 30. What effect does the number in front of x Sine Graph y = sinxo have on the y = sin2xoS4 Credit graphs ? y = sin4xo 3 y = sin0.5xo www.mathsrevision.com 2 1 0 90o 180o 270o 360o -1 -2 -3 created by Mr. Lafferty
  • 31. Trigonometry GraphsS4 Credit y = a sin (bx) www.mathsrevision.com How many times it repeats itself in 360o For a > 1 stretches graph in the y-axis direction For a < 1 compresses graph in the y - axis direction For a - negative flips graph in the x – axis. created by Mr. Lafferty
  • 32. Cosinecosx y= o y = cos2xoS4 Credit y = cos3xo 3 www.mathsrevision.com 2 1 0 90o 180o 270o 360o -1 -2 -3 created by Mr. Lafferty
  • 33. Trigonometry GraphsS4 Credit y = a cos (bx) www.mathsrevision.com How many times it repeats itself in 360o For a > 1 stretches graph in the y-axis direction For a < 1 compresses graph in the y - axis direction For a - negative flips graph in the x – axis. created by Mr. Lafferty
  • 34. Trigonometry GraphsS4 Credit y = a tan (bx) www.mathsrevision.com How many times it repeats itself in 180o For a > 1 stretches graph in the y-axis direction For a < 1 compresses graph in the y - axis direction For a - negative flips graph in the x – axis. created by Mr. Lafferty
  • 35. Write down the equations for the graphs shown ? Trig Graph y = 0.5sin2xo y = 2sin4xoS4 Credit Combinations y = -3sin0.5xo 3 www.mathsrevision.com 2 1 0 90o 180o 270o 360o -1 -2 -3 created by Mr. Lafferty
  • 36. Write down y = 1.5cos2xo equations for the graphs shown? Cosine y = -2cos2xo CombinationsS4 Credit y = 0.5cos4xo 3 www.mathsrevision.com 2 1 0 90o 180o 270o 360o -1 -2 -3 created by Mr. Lafferty
  • 37. Combination GraphsS4 Credit www.mathsrevision.com Now Try MIA Ch11 Ex 5.1 Page 227 created by Mr. Lafferty
  • 38. C moves the graph Trigonometry Graphs = 360 up or down in the Period o y-axis direction bS4 Credit y = a sin (bx) + c www.mathsrevision.com How many times it repeats a - Amplitude itself in 360o For a > 1 stretches graph in the y-axis direction For a < 1 compresses graph in the y - axis direction For a - negative flips graph in the x – axis. created by Mr. Lafferty
  • 39. Sine GraphS4 Credit Simply move graph up by 1 www.mathsrevision.com 1 0 45o 90o 180o 270o 360o Given the basic y = sin x -1 graph what does the graph of y = sin x + 1 look like? created by Mr. Lafferty
  • 40. Cosine Graph Given the y = cos x graph. What does the graph of y = cos x – 0.5 look like?S4 Credit Simply move down by 0.5 www.mathsrevision.com 1 0 90o 160 180o o 270o 360o -1 created by Mr. Lafferty
  • 41. Write down equations for graphs shown ? Trig Graph= 0.5sin2x y o + 0.5 Combinationsy = 2sin4x - 1 oS4 Credit 3 www.mathsrevision.com 2 1 0 90o 180o 270o 360o -1 -2 -3 created by Mr. Lafferty
  • 42. Write down equations for the graphs shown? Cosine y = cos2xo + 1 Combinationsy = -2cos2x - 1 oS4 Credit 3 www.mathsrevision.com 2 1 0 90o 180o 270o 360o -1 -2 -3 created by Mr. Lafferty
  • 43. Combination GraphsS4 Credit www.mathsrevision.com Now try MIA Ch11 Ex 5.2 Page 231 created by Mr. Lafferty
  • 44. StarterS4 Credit 1. Make b the subject of the formula www.mathsrevision.com c=b+d 2. Use the quadratic formula to solve x + 6x + 2 2 3. Sketch the function y = 2sin4x created by Mr. Lafferty
  • 45. Solving Trig EquationsS4 Credit Learning Intention Success Criteria www.mathsrevision.com 1. To explain how to solve 1. Use the rule for solving any trig equations of the form ‘ normal ‘ equation a sin xo + 1 = 0 2. Realise that there are many solutions to trig equations depending on domain. created by Mr. Lafferty
  • 46. Solving Trig EquationsS4 Credit www.mathsrevision.com Sin +ve All +ve 180o - xo 180o + xo 360o - xo Tan +ve Cos +ve 1 2 3 4 created by Mr. Lafferty
  • 47. Solving Trig Equations what GraphicallyS4 Credit a sin xo + b = 0 we trying to are solve Example 1 : www.mathsrevision.com Solving the equation sin xo = 0.5 in the range 0o to 360o sin xo = (0.5) xo = sin-1(0.5) xo = 30o There is another solution xo = 150o 1 2 3 4 created by Mr. Lafferty (180o – 30o = 150o)
  • 48. Solving Trig Equations what GraphicallyS4 Credit a sin xo + b = 0 we trying to are solve Example 1 : www.mathsrevision.com Solving the equation 3sin xo + 1= 0 in the range 0o to 360o sin xo = -1/3 Calculate first Quad value xo = 19.5o x = 180o + 19.5o = 199.5o There is another solution 1 2 3 4 ( 360o - 19.5o = 340.5o) created by Mr. Lafferty
  • 49. Solving Trig Equations what GraphicallyS4 Credit a cos xo + b are we trying to =0 solve Example 1 : www.mathsrevision.com Solving the equation cos xo = 0.625 in the range 0o to 360o cos xo = 0.625 xo = cos -1 0.625 xo = 51.3o There is another solution (360o - 53.1o = 308.7o) 1 2 3 4 created by Mr. Lafferty
  • 50. Solving Trig Equations what GraphicallyS4 Credit a tan xo + b are we trying to =0 solve Example 1 : www.mathsrevision.com Solving the equation tan xo = 2 in the range 0o to 360o tan xo = 2 xo = tan -1(2) xo = 63.4o There is another solution x = 180o + 63.4o = 243.4o 1 2 3 4 created by Mr. Lafferty
  • 51. Solving Trig EquationsS4 Credit www.mathsrevision.com Now try MIA Ch11 Ex6.1, 6.2 and 7.1 (page 236) created by Mr. Lafferty
  • 52. StarterS4 Credit 1. Make a the subject of the formula www.mathsrevision.com 5 = 10b + a 2. Use the quadratic formula to solve x + 5x + 1 2 3. Sketch the function y = 4sin3x created by Mr. Lafferty
  • 53. Solving Trig EquationsS4 Credit Learning Intention Success Criteria www.mathsrevision.com 1. To explain some special 1. Know and learn the two trig relationships special trig relationships. sin 2 xo + cos 2 xo = ? 2. Apply them to solve and problems. tan xo and sin x cos x created by Mr. Lafferty
  • 54. Solving Trig EquationsS4 Credit Lets investigate www.mathsrevision.com sin 2xo + cos 2 xo = ? Calculate value for x = 10, 20, 50, 250 sin 2xo + cos 2 xo = 1 Learn ! created by Mr. Lafferty
  • 55. Solving Trig EquationsS4 Credit Lets investigate www.mathsrevision.com sin xo tan xo and cos xo Calculate value for x = 10, 20, 50, 250 sin xo tan xo = cos xo Learn ! created by Mr. Lafferty
  • 56. Solving Trig EquationsS4 Credit www.mathsrevision.com Now try MIA Ex8.1 Ch11 (page 238) created by Mr. Lafferty