The Quadratic function f(x) = x 2 2x 3. y y = x 2 2x 3. We will now begin to study the graphs of the trig functions, y = sinx, y = cosx and y = tanx.

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1 Chapter 7 Trigonometric Graphs Introduction We have alread looked at the graphs of various functions : The Linear function f() = The Quadratic function f() = The Hperbolic function f() = = = = We will now begin to stud the graphs of the trig functions, = sin, = cos and = tan. The Sine Function ( = sin ) There are computer or graphics packages that could assist here. You are going to look up the sine of various angles from to and plot them on a graph Cop this table and use our calculator to complete it (to decimal places each time) sin Now take a sheet of A two millimetre graph paper and use it in the landscape position. Plot our points from the table and join them up with a smooth curve Chapter 7 this is page 7 Trig Graphs

2 Your graph should have ended up looking like this :- = sin = sin Stud it carefull Eercise 7. Practice sketching the graph several times on a smaller scale like this :- note the smooth wav shape. What is the highest value the graph attains? (c) What is its lowest value? (d) For what values of does the curve cut the -ais? Make a new neat small sketch of = sin, but this time etend the -ais to go from to +7. (see below) Use our calculator to find sin, sin, sin and sin 7, plot these points on our diagram and sketch the net bit of the sine graph. (c) Repeat for sin ( 9 ), sin ( 8 ), sin ( 7 ) and sin ( ) and draw this bit of the sine graph. (d) For the graph of = sin, state the maimum and minimum values. (how high and low it goes). (e) The vertical distance between the maimum and minimum values is called the amplitude. What is the amplitude of the sine graph? (f) The horizontal distance between points on the graph where the pattern repeats itself is called the period of the graph. What is the period of the sine graph? Chapter 7 this is page 7 Trig Graphs

3 The Cosine Function ( = cos ) You are going to look up the cosine of various angles from to and plot them on a graph. Cop this table and use our calculator to complete it (to decimal places each time) cos Now take a sheet of A two millimetre graph paper and use it in the landscape position. Plot our points from the table and join them up with a smooth curve Show our finished smooth cosine graph to our teacher. Eercise 7. Practice sketching the graph several times on a smaller scale like this :- note again the smooth wav shape. What is the highest value the graph attains? (c) What is its lowest value? (d) For what values of does the curve cut the -ais? Make a new neat sketch of = cos, and etend the -ais to go from to Chapter 7 this is page 7 Trig Graphs

4 Use our calculator to find cos, cos, cos and cos 7, plot these points on our diagram and sketch the net bit of the cosine graph. (c) Repeat for cos ( 9 ), cos ( 8 ), cos ( 7 ) and cos ( ) and draw this bit of the cosine graph. (d) For the graph of = cos, state the maimum and minimum values. (how high and low it goes). (e) The vertical distance between the maimum and minimum values is called the amplitude. What is the amplitude of the cosine graph? (f) The horizontal distance between points on the graph where the pattern repeats itself is called the period of the graph. What is the period of the cosine graph?. Without looking at the last page, make a quick sketch of = sin and = cos, marking in the important values on both the and the -aes. Write a couple of sentences describing both graphs - in what was are the similar and in what was are the different? (shape, maimum/minimum values, amplitudes, periods)? note :- when sketching an sine or cosine graph, it is easier if ou do so in the following order :- step draw the aes first. step then draw the sine, (or cosine), shaped graph. step finall fill in the scales on the aes Tr to remember to draw our trig graphs in the above order. We will, through the course of the net pages, draw graphs like :- = sin = cos = sin = cos = sin = cos = sin + = cos but let us first of all look at the tangent graph. Chapter 7 this is page 7 Trig Graphs

5 The Tangent Function ( = tan ) The tangent graph looks totall different from the sine and cosine graphs. Cop this table and use our calculator to complete it (to decimal places each time). (c) tan ? At 9, we sa the tangent is undefined (it is too large a number to find - infinit!) Now take a sheet of A two millimetre graph paper (or half-cm paper) and use it in the portrait position. Plot the 7 points from the table and join them up with a smooth curve. Note the shape - the slope rising ver slowl at first, then accelerating towards infinit! Etend our table to show the -values from 9 to tan Etend our -ais, plot these points and join them to show the second half of the smooth tangent curve. Show our finished smooth tangent graph to our teacher. Eercise 7. Practice sketching the graph several times on a smaller scale like this :- Is there a highest value the graph attains? (c) Is there a lowest value? (d) For what values of does the curve cut the -ais?. Make a new neat sketch of = tan { 8} and etend the -ais to go from 8 to +. What is the period of the tangent function? The tangent function is not as important as the sine and cosine functions. It does not appear often in real life Chapter 7 this is page 7 Trig Graphs

6 Other Sine and Cosine Functions ( = asin and = acos ) = sin You are going to draw the graph of = sin b looking up various values of. For eample, when = => sin = => sin = = Cop this table and use our calculator to complete it (to decimal places each time) sin Draw a set of aes on squared paper, plot the above points and join them up with a smooth curve. Show our graph of :- = sin to our teacher Eercise 7. This time, just sketch the graph = sin on a smaller scale like this :- What is the highest value the graph attains? (c) What is its lowest value? (d) What is the period of = sin? (e) For what values of does the curve cut the -ais? You should have noticed the following :- the graph is identical in shape to that of = sin. its maimum and minimum values are now + and, and its amplitude becomes. its period is still it is not altered b the sin. You will find :- the maimum/minimum value of = sin is + and. Its period is still the maimum/minimum value of = sin is + and. Its period is still the maimum/minimum value of = cos is + and. Its period is still the maimum/minimum value of = sin is + and. Its period is still Chapter 7 this is page 7 Trig Graphs

7 . This time, ou are going to sketch the graph of = sin. step step step draw a set of and -aes. draw the sine-shaped curve. now fill in the to and +/ What is the maimum value and what is the minimum value the graph attains? (c) What is the amplitude and period of = sin?. Sketch the graph of = cos, { }. remember - aes first, then cosine-shaped graph and lastl the scales. 7. Each of the following trig graphs represents a function of the form = asin or = acos. Write down the equation of each function. What are the maimum-minimum values? (c) What is the amplitude and the period of = cos?. Make neat sketches of the following trig. graphs, using the values - { }. = 8sin. = sin. (c) = cos. (d) = cos. (e) = sin. (f) = tan.. This time ou are going to draw the graph of the function = sin. Remember that this is a sketch of = sin. How do ou think = sin will differ? What are the maimum-minimum values? (c) What is the amplitude and the period of = sin?. Make a neat labelled sketch of :- = cos = sin (c) (d) Chapter 7 this is page 7 Trig Graphs

8 More Sine and Cosine Functions ( = sina and = cosa ) = sin We are going to stud the = sin for various values of. for eample, when = => = => sin = sin = 87 Cop this table and use our calculator to complete it (to decimal places each time). 9 8 sin Draw a set of aes on squared paper, plot the above nine points and join them up with a smooth curve. Show our graph of = sin to our teacher. 9 8 Eercise 7. This time, just sketch the graph = sin on a smaller scale like this :- What is the highest value the graph attains? (c) What is its lowest value? (d) What is the amplitude and the period of = sin? (e) For what values of does the curve cut the -ais? 9 8 You should have noticed the following :- the graph is identical in shape to that of = sin. its maimum and minimum values are still + and, and its amplitude is still. its period is no longer its period is now = 8. You will find that :- the period of = sin has a period of = the period of = sin has a period of = the period of = cos has a period of = 9 the period of = tan has a period of 8 = 9 Chapter 7 this is page 77 Trig Graphs

9 . This time, ou are going to sketch the graph of = sin. step step step draw a set of and -aes. draw the sine-shaped curve. to find the period :- divide = 9. 9 What is the maimum value and what is the minimum value of the function? (c) What is the period of = sin?. Sketch the graph of = cos. remember - aes first, then cosine-shaped graph and lastl the scales. 7. Each of the following trig graphs represents a function of the form = asinb or = acosb. Write down the equation of each function. What are the maimum-minimum values? (c) What is the period of = cos?. Make neat sketches of the following trig. graphs. = sin. = sin. (c) = cos. (d) = 7cos. (e) = sin. (f) = tan. (careful) 9 8. This time ou are going to draw the graph of the function = 9sin. Remember that this is a sketch of = 9sin How do ou think = 9sin will differ? What are the maimum-minimum values? (c) What is the period and amplitude of = 9sin? (c) 7 (d). Make a neat labelled sketch of ccle of :- = cos = sin (c) = sin (d) = 8 cos 8 7 Chapter 7 this is page 78 Trig Graphs

10 Yet more Sine and Cosine Functions ( = asin + b and = acos + b) When the sine or cosine function has a number added on (or subtracted), the simple effect is to slide the basic sine or cosine function upwards (or downwards) b that amount. Eample :- = sin + Eample :- = cos = sin + 9 = cos = sin = cos Note :- The period of the new function with the added (or subtracted) term remains the same. The amplitude stas the same (the difference between the highest and lowest points). But the maimum and the minimum values change. In eample, the maimum and minimum changes from and > to and. In eample, the maimum and minimum changes from and > to and 9. To draw = sin +, simpl sketch the graph of = sin and move each point up b. Eercise 7. Shown below is a sketch of the function = sin. Make a neat cop of the graph, showing the graph dotted as in the sketch On our drawing, show also the graph of the function :- = sin.. Make a neat (dotted) sketch of the function = cos, showing all the main features and values. On the same graph, show the function :- = cos +.. This time, make a sketch showing = sin. On the same graph, show the function :- = sin, showing all its main features.. This time, sketch the graph of the function :- = cos, On the same graph, show = cos +, indicating all the main features and values. Chapter 7 this is page 79 Trig Graphs

11 . Make neat sketches of each of the following, showing all the main features and values. (hint :- sketch the basic trig function first). = sin + = cos (c) = sin (d) = cos. Shown below is the graph of = sin. 7 (remember how it is upside-down ). Make a neat sketch of this (dotted) curve, and show on it the graph of = sin Sketch the graph of = cos, (dotted), and show also the graph of = cos, indicating all of its main features and values. 8. Make neat sketches of each of the following, showing all the main features and values. = sin + = cos (c) = sin (d) = cos 9. Work out the equation of the following trig function from its graph :-. Determine the equation of this trig graph : Determine the equation of each of the following trig functions from their graphs :- 8 (c) (e) (d). Harder - Each of these functions is of the form :- (f) = asinb + c. Determine the equation of each. (c) (d) Chapter 7 this is page 8 Trig Graphs

12 . The pedal on this biccle crank is rotated. Its height above (and below) the centre of the shank is noted as the pedal rotates. This is shown on the graph below. h. A chalk-mark is made on the tre of a biccle wheel. As the wheel rotates along the ground, its height in centimetres is recorded and shown on the graph below. h h h 9 t t t = time in seconds t represents the time (in seconds) h represents the height (in centimetres). What is the period of the graph? (this is the time taken for rotation). Write down the equation of the graph :- h =... sin...t.. The water level rises and falls ever hours in a harbour as the tide comes in and out. From the graph, sa what the diameter of the wheel must be. What is the period of the graph? (c) Write down the equation of the graph :- h =... sin...t The graph below shows the average number of hours of dalight, (each da), there is throughout the ear starting from the month of June. N P d m 7 7 A graph, showing the depth (d m) of water measured from point P, is shown below. The time (t) is measured in hours. d How high above P is the water at high tide? Write down the equation of the graph :- d =... sin...t. t t (June) N is the number of hours sunshine dail t is the number of months after June. What is the maimum number of hours of sunshine each da, and in which month? What is the minimum number, and when? (c) The equinoes (Spring and Autumn) are when there is an equal number of hours of light and dark. Which months? (d) Find the equation of the graph in the form :- N =... cos...t Chapter 7 this is page 8 Trig Graphs

13 Remember Remember...? Topic in a Nutshell. Sketch the sine graph, the cosine graph and the tangent graph, { }, on different diagrams, indicating the shape of each and all the important points through which the pass.. Write down the equations of the following graphs :- (c) Write down the equations of the following graphs :- (c) Write down the equations of the following graphs :- (c) 8. Write down the equations of the following graphs :- (c) Make a neat sketch of each function, showing the shape, scale and important points on our graphs. = sin = cos (c) = sin (d) = 8cos (e) = sin + (f) = sin (g) = cos + (h) = sin (i) = cos Chapter 7 this is page 8 Trig Graphs