Trig Graphs. What is a Trig graph? This is the graph of a trigonometrical function e.g.

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Stewart Dorsey
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1 Trig Graphs What is a Trig graph? This is the graph of a trigonometrical function e.g. sin, cos or tan How do we draw one? We make a table of value using the calculator. Tr to complete the one below (work to decimal places). (degrees) sin (degrees) sin (degrees) sin (degrees) sin Below are the value for ou to check: (degrees) sin (degrees) sin (degrees) sin (degrees) sin On the net sheet, we will plot the above values on a graph

2 Graph of sin sin() If we continue past 60, then the graph repeats itself as below: You can check this with our calculator, b calculating sin 450 etc and making sure it agrees with the graph. sin() It should be noted that: The maimum value of The minimum value of sin is + sin is The graph is centred on the line 0 The graph repeats itself ever 60 there is a one complete wave in 60

3 The graph of cos We can repeat the previous steps. (degrees) cos (degrees) cos (degrees) cos (degrees) cos Below are the value for ou to check: (degrees) cos (degrees) cos (degrees) cos (degrees) cos On the net sheet, we will draw the graph. For clarit on the graph, the plotted points from above will not be visible. However, ou should check to see if ou agree that the graph matches the points above. Check one or two values to be sure.

4 Graph of cos cos() If we continue past 60, then the graph repeats itself as below: You can check this with our calculator, b calculating cos 450 etc and making sure it agrees with the graph. cos() It should be noted that: The maimum value of The minimum value of cos is + cos is The graph is centred on the line 0 The graph repeats itself ever 60 there is a one complete wave in 60

5 Graph of tan Similarl, we can draw this graph also. The results are shown below. 5 tan() If we continue past 60, then the graph repeats itself as below: You can check this with our calculator, b calculating tan 540 etc and making sure it agrees with the graph. 5 tan() It should be noted that: There is no maimum or minimum value of tan The graph is centred on the line 0 The graph repeats itself ever 80

6 Amplitude We can use the graphs to make observations about trig functions. sin maimum minimum sin maimum minimum sin maimum - minimum In general, for the graph asin the maimum value will be a the minimum value will be a Where a is the distance of the top of the wave from the centre line. We call a the amplitude of the wave (how big it is). 5 amplitude This same definition applies to the cosine wave. We shall not be concerned ver much with the tangent waveform at Standard Grade.

7 Periodicit We can use the graphs to make observations about trig functions. sin One waveform in 60 - sin Two waveforms in 60 - sin Three waveforms in 60 - In general, for the graph sin b there will be b complete waves in 60 We sa that the period of the waveform, is the number of degrees for one complete wave. The period will be given b: 60 b We determine b b determining how man complete waves there are in 60 The equation of this graph is: sin because there are 4 complete waveforms in complete waves in 60 This also applies to the cosine wave.

8 Inversion reflected or negative graphs We can use the graphs to make observations about trig functions sin - - sin 60 - Note the reflection in the ais - 60 cos - - cos 60 - Note the reflection in the ais -

9 We are now in a position to be able to write down the equation of a sine or cosine graph simpl b looking for the amplitude and the periodicit. e.g. The graph is of the form asin b Amplitude is There are waveforms in 60 - Equation is: sin - e.g. The graph is of the form asin b 4 Amplitude is There are 4 waveforms in 60 Equation is: sin e.g. The graph is of the form a cosb Amplitude is There are waveforms in 60 Equation is: cos e.g. The graph is of the form a cosb Amplitude is There is onl ½ a waveform in 60 Equation is: cos e.g. The graph is of the form asin b Amplitude is There is onl ¼ a waveform in 60 Equation is: cos

10 e.g. The graph is of the form asin b (45, ) Amplitude is ¼ of the wave is in 45, whole wave in 80 So, two waves in 60 Equation is: sin e.g. The graph is of the form a cosb (0, 4) Amplitude is 4 ¼ of the wave is in 0, whole wave in 0 So, three waves in 60 Equation is: 4sin e.g. The graph is of the form a cosb (60, ) Amplitude is, note this is: cos wave ½ of the wave is in 60, whole wave in 0 So, three waves in 60 Equation is: cos e.g. The graph is of the form asin b Amplitude is, note this is: sin wave ¼ of the wave is in 45, whole wave in 80 So, two waves in 60 Equation is: sin (45, )

11 Some past paper questions:. Shown is the graph of a sin b Write down the values of a and b. note sin wave; Amplitude 5; ¼ of wave in 0, whole wave in 0, waves in 60 a 5, b. On a certain da the depth, D metres, of water at a fishing port, t hours after midnight, is given b the formula D sin(0 t) a) Find the depth of water at.0 pm b) The depth of water in the harbour is recorded each hour. What is the maimum difference in the depths of water in the harbour, over the 4 hour period? Show clearl all our working. a).0 pm is.5 hours after midnight D sin(0.5) sin(405) metres. b) The maimum value of sin is ; so ma value of D is metres The minimum value of sin is ; so min value of D is metres. Hence maimum difference in depths of water 9 metres. 4. The diagram shows the graph of k sin a, 0 60 Find the values of a and k. Amplitude ; waves in 60 a, k 5. The diagram shows the graph of a cos b, 0 60 Find the values of a and b. Amplitude ; waves in 60 a, b

Section 7.1 Graphs of Sine and Cosine

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