1 / 27

Graphs of Trigonometric Functions

Graphs of Trigonometric Functions. Graphs of Trigonometric Functions. This chapter focuses on using graphs of sin θ , cos θ and tan θ We will be seeing how to work out values of these from the graphs We are also going to look at transformations of these graphs.

iliana-briggs
Download Presentation

Graphs of Trigonometric Functions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Graphs of Trigonometric Functions

  2. Graphs of Trigonometric Functions • This chapter focuses on using graphs of sinθ, cosθ and tanθ • We will be seeing how to work out values of these from the graphs • We are also going to look at transformations of these graphs

  3. I teach using the Trig graphs so haven’t included sections 8A and 8B on CAST diagrams!

  4. Teachings for Exercise 8C

  5. -90º -360º -180º -270º Graphs of Trigonometric Functions y y = sinθ 1 You need to be able to recognise the graphs of sinθ, cosθ and tanθ You will have seen all these graphs on your GCSE The key points to remember are the peaks/troughs of each, and the points of intersection The Cos graph is the same as the Sin graph, but shifted along (it starts at 1 instead of 0) The Tan graph has lines called asymptotes. These are points the graph approaches but never reaches (90º, 270º etc…) θ 0 360º 90º 180º 270º -1 y y = cosθ 1 θ 0 -180º -90º -360º -270º 360º 90º 180º 270º -1 y = tanθ 1 θ 0 -180º -90º 90º -360º -270º 360º 180º 270º -1 Period (length of wave) = 360º for Sin and Cos, and 180º for Tan 8C

  6. Graphs of Trigonometric Functions y y = sinθ 1 You need to be able to recognise the graphs of sinθ, cosθ and tanθ These are the same graphs, but with radians instead… θ 0 -π 2 π 2 π -π 3π 2 -90º -2π -3π 2 -360º 360º -180º 90º 180º 270º -270º 2π -1 y y = cosθ 1 θ 0 -π 2 π 2 π -π 3π 2 -90º -2π -3π 2 -360º 360º -180º 90º 180º 270º -270º 2π -1 y = tanθ 1 θ 0 -π 2 π 2 π -π 3π 2 -90º -2π -3π 2 -360º 360º -180º 90º 180º 270º -270º 2π -1 8C

  7. Graphs of Trigonometric Functions y = sinθ y You need to be able to recognise the graphs of sinθ, cosθ and tanθ You need to be able to work out larger values of sin, cos and tan as acute angles (0º - 90º)  Write sin 130º as sine of an acute angle (sometimes asked as a ‘trigonometric ratio’) Sin 130º = Sin 50º -40 -40 1 50 130 θ 0 360º 90º 180º 270º -1  Draw a sketch of the graph  Mark on 130º  Using the fact that the graph has symmetry, find an acute value of θ which has the same value as sin 130 8C

  8. Graphs of Trigonometric Functions y = cosθ y You need to be able to recognise the graphs of sinθ, cosθ and tanθ You need to be able to work out larger values of sin, cos and tan as acute angles (0º - 90º)  Write cos (-120)º as cos of an acute angle Cos(-120)º = -Cos 60º +60 +60 1 -60 60 +30 θ 0 +30 -90º -180º 90º 180º 270º -270º -120 -1  Draw a sketch of the graph  Mark on -120º  Using the fact that the graph has symmetry, find an acute value of θ which has the same numerical value as cos (-120)  The value you find here will have the same digits in it, but will be multiplied by -1 8C

  9. 1 3 1 3 + + Graphs of Trigonometric Functions y = tanθ You need to be able to recognise the graphs of sinθ, cosθ and tanθ You need to be able to work out larger values of sin, cos and tan as acute angles (0º - 90º)  Write tan 4π/3 as tan of an acute angle Tan 4π/3 = Tan π/3 1 1 3 4 3 θ 0 π 2 π 3π 2 2π -1  Draw a sketch of the graph  Mark on 4π/3  Using the fact that the graph has symmetry, find an acute value of θ which has the same numerical value as tan 4π/3 8C

  10. Teachings for Exercise 8D

  11. Graphs of Trigonometric Functions You need to be able to find the exact values of some Trigonometrical Ratios Some values of Sin, Cos or Tan can be written using fractions, surds, or combinations of both… We can use an Equilateral Triangle with sides of length 2 to show this. Using Pythagoras, the missing side in the right angled triangle is √3 (Square root of 22-12) 60˚ 2 2 60˚ 60˚ 2 Hyp 30˚ 2 Opp √3 Opp Hyp 1 2 Sinθ = Sin30 = 60˚ √3 2 Sin60 = 1 Opp 8D

  12. Graphs of Trigonometric Functions You need to be able to find the exact values of some Trigonometrical Ratios Some values of Sin, Cos or Tan can be written using fractions, surds, or combinations of both… We can use an Equilateral Triangle with sides of length 2 to show this. Using Pythagoras, the missing side in the right angled triangle is √3 (Square root of 22-12) 60˚ 2 2 60˚ 60˚ 2 Hyp 30˚ 2 Adj √3 Adj Hyp √3 2 Cosθ = Cos30 = 60˚ 1 2 Cos60 = 1 Adj 8D

  13. Graphs of Trigonometric Functions You need to be able to find the exact values of some Trigonometrical Ratios Some values of Sin, Cos or Tan can be written using fractions, surds, or combinations of both… We can use an Equilateral Triangle with sides of length 2 to show this. Using Pythagoras, the missing side in the right angled triangle is √3 (Square root of 22-12) 60˚ 2 2 60˚ 60˚ 2 30˚ 2 Adj Opp √3 Opp Adj 1 √3 √3 3 = Tanθ = Tan30 = 60˚ Tan60 = √3 1 Adj Opp 8D

  14. Graphs of Trigonometric Functions You need to be able to find the exact values of some Trigonometrical Ratios Some values of Sin, Cos or Tan can be written using fractions, surds, or combinations of both… We can also do a similar demonstration with a right-angled Isosceles triangle, with the equal sides being of length 1 unit. Using Pythagoras’ Theorem, the hypotenuse will be of length √2 (Square root of 12 + 12) Hyp √2 Opp 1 45˚ 1 Opp Hyp 1 √2 √2 2 = Sinθ = Sin45 = 8D

  15. Graphs of Trigonometric Functions You need to be able to find the exact values of some Trigonometrical Ratios Some values of Sin, Cos or Tan can be written using fractions, surds, or combinations of both… We can also do a similar demonstration with a right-angled Isosceles triangle, with the equal sides being of length 1 unit. Using Pythagoras’ Theorem, the hypotenuse will be of length √2 (Square root of 12 + 12) Hyp √2 1 45˚ 1 Adj Adj Hyp 1 √2 √2 2 = Cosθ = Cos45 = 8D

  16. Graphs of Trigonometric Functions You need to be able to find the exact values of some Trigonometrical Ratios Some values of Sin, Cos or Tan can be written using fractions, surds, or combinations of both… We can also do a similar demonstration with a right-angled Isosceles triangle, with the equal sides being of length 1 unit. Using Pythagoras’ Theorem, the hypotenuse will be of length √2 (Square root of 12 + 12) √2 Opp 1 45˚ 1 Adj Opp Adj 1 1 = 1 Tanθ = Tan45 = 8D

  17. 8E is just a reminder of the Trig Graph shapes!

  18. Teachings for Exercise 8F

  19. Graphs of Trigonometric Functions y y = sinθ You need to be able to recognise transformations of graphs, and sketch them Transformation type 1 This stretches the graph vertically by a factor ‘a’. “Multiplying sinθ by a number will affect the y value directly” 1 θ 0 360º 90º 180º 270º -1 y 3 y = 3sinθ θ 0 360º 90º 180º 270º Y values 3 times as big -3 y = ½sinθ Y values halved y 0.5 θ 0 360º 90º 180º 270º -0.5 8F

  20. Graphs of Trigonometric Functions y y = sinθ You need to be able to recognise transformations of graphs, and sketch them Transformation type 1 This stretches the graph vertically by a factor ‘a’. 1 θ 0 360º 90º 180º 270º -1 y y = -sinθ 1 θ 0 360º 90º 180º 270º Reflection in the x axis -1 (all the y values will ‘swap sign’) y y = sin(-θ) 1 Reflection in the y axis θ 0 360º 90º 180º 270º (You get the same y values for the reversed x value. -90 gives the result 90 would have) -1 8F

  21. Graphs of Trigonometric Functions y y = cosθ You need to be able to recognise transformations of graphs, and sketch them Transformation type 1 This stretches the graph vertically by a factor ‘a’. 1 θ 0 360º 90º 180º 270º -1 y y = -cosθ 1 θ 0 360º 90º 180º 270º Reflection in the x axis -1 y y = cos(-θ) 1 Reflection in the y axis θ 0 360º 90º 180º 270º -1 8F

  22. Graphs of Trigonometric Functions y y = sinθ You need to be able to recognise transformations of graphs, and sketch them Transformation type 2 This shifts the graph vertically ‘a’ units.  It is important to note that the ‘a’ is added on AFTER doing ‘sinθ’ “Adding an amount onto sinθ is a vertical shift” 1 θ 0 360º 90º 180º 270º -1 y y = sinθ + 1 1 θ 0 360º 90º 180º 270º -1 Y values all increase by 1 y y = -2 + sinθ -1 Y values all decrease by 2 θ -2 -3 8F

  23. Graphs of Trigonometric Functions y y = sinθ You need to be able to recognise transformations of graphs, and sketch them Transformation type 3 This shifts the graph horizontally ‘-a’ units. NOTE: The ‘a’ is added to θ before we work out the sine value… “Adding/Subtracting an amount from the bracket is a horizontal shift” 1 θ 0 360º 90º 180º 270º -1 90 y y = sin(θ + 90) 1 θ 0 360º 90º 180º 270º -1 Y takes the same set of values, for values of θ that are 90 less than before y 30 y = sin(θ – 30) 1 Y takes the same set of values, for values of θ that are 30 more than before θ 0 360º 90º 180º 270º -1 8F

  24. Graphs of Trigonometric Functions y y = sinθ You need to be able to recognise transformations of graphs, and sketch them Transformation type 4 This stretches the graph horizontally by a factor ‘1/a’ “Multiplying or dividing θ in the bracket is a horizontal stretch/squash” 1 θ 0 360º 90º 180º 270º -1 y y = sin2θ 1 θ 0 90º 180º 270º 360º Same set of Y values, for half the θ values -1 y y = sin(θ/3) Same set of y values, for triple the θ values 1 θ 0 1080º 270º 540º 810º -1 8F

  25. Graphs of Trigonometric Functions (90, 1.5) y y = sinθ + k You need to be able to answer questions with unknowns in The graph shows the Function: f(x) = Sinθ + k a) Write down the value of k  0.5 (Graph 0.5 units higher) b) What is the smallest positive value of θ that gives a minimum point?  270˚ c) What is the value of Sinθ at this point?  -0.5 1 θ 0 360º 90º 180º 270º -1 8F

  26. Graphs of Trigonometric Functions y y = cos(θ+k) You need to be able to answer questions with unknowns in The graph shows the Function: f(x) = Cos(θ + k) a) Write down the value of k  20 (Graph moved 20 units left)  f(x) = Cos(θ + 20) b) What is the value of θ at x?  x = 250˚ c) What are the coordinates of the minimum?  (160, -1) d) What is the value of Cosθ at y? 1 y 70º 250º xº θ 0 -1 f(x) = Cos(θ + k) We know k f(x) = Cos(θ + 20) On the y axis, θ = 0. f(x) = Cos(20) Work out the answer! f(x) = 0.94 (2dp) 8F

  27. Summary • We have been reminded of the graphs for sine, cosine and tan • We have looked at finding equivalent values on these graphs • We have also looked at various graph transformations

More Related