Graphs of Trigonometric Function

The graph of the followings will be considered

(a) y = sinѳ, 0◦ ≤ ѳ ≤ 360◦

(b) y = cosѳ, 0◦ ≤ ѳ ≤ 360◦

(c) y = tanѳ, 0◦ ≤ ѳ ≤ 360◦

The graph of y = sinѳ, ѳ◦ ≤ ѳ ≤ 360◦

On the graph sheet, draw a long horizontal axis in the middle. Mark a point 0’, 3cm to the left of the origin 0. With centre 0’ draw a circle of radius 2cm. Draw a vertical axis through 0. Call the horizontal axis ѳ – axis and the vertical axis y – axis.

On the ѳ – axis choose a scale of 2cm to 1 unit. Using your protractor, mark the angles: 0◦, 30◦, 60◦, 90◦, 120◦, 150◦ … 330◦ as shown in

Draw a horizontal line through 30◦ on the circle. Draw a vertical line through 30◦ on the circle. Draw a vertical line through 30◦ on the ѳ – axis to meet the horizontal line. Mark the point of intersection of these two lines with a small neat cross. Repeat the above procedure for the angles 60◦, 90◦, 120◦, … 330◦. You will obtain a series of points. Join the points by a smooth curve. The curve you obtain is the graph of y = sinѳ.

Essential features of the graph of y = sinѳ:

(a) The graph of y = sinѳ forms a wave – like pattern. It is said to oscillate.

(b) The maximum value of y = sinѳ is 1 and it occurs when ѳ = 90◦.

(c) The minimum value of y = sinѳ is -1 it occurs when ѳ = 270◦.

(d) The graph repeats itself at intervals of 360◦. The sine function is an example of a periodic function because it repeats itself at intervals of 360◦. The function is said to have a periodicity of 360◦.

(e) The length AE on the graph is called the amplitude of the function.

The Graph of y = cosѳ, ѳ◦ ≤ ѳ ≤ 360◦

The graph of y = cosѳ can be drawn in a manner similar to that of y = sinѳ except that the angles are measured from OR in the clockwise sense as shown in Fig. 14.17. This is so because cosѳ = sin(90◦ – ѳ).

Essential features of the graph of y = cosѳ:

(a) All the essential features that hold for the graph of y = sinѳ also hold for the graph of y = cosѳ.

(b) In addition, the cosine curve lags behind the sine curve by a difference of 90◦. The difference is usually called a Phase difference. In other words, the cosine curve lags behind the sine curve by a phase difference of 90◦.

(c) Both the sine curve and the cosine curve demonstrate some physical phenomena like tidal waves, sound waves alternating currents, e.t.c.

The Graph of y = tanѳ, ѳ◦ ≤ 360◦

The graph of y = tanѳ is easier to draw using a table of values than using projections from a unit circle. Make a table of values of y = tanѳ from 0◦ to 360◦ as shown in Table 14.2

Essential features of the tangent curve:

(a) The curve consists of three parts between 0◦ and 360◦.

(b) Since the tangent function is not defined at 90◦ and 270◦, the function is said to be discontinuous at these points.(c) The curve rises and falls rapidly at anglesvery close to 90◦ and 270◦ respectively. The curve approaches the vertical lines at 90◦ and 270◦ but never touches them. These vertical lines at 90◦ and 270◦ ate called Asymptotes. The asymptotes are shown by dotted lines.

(d) The tangent function is also a periodic function. It has a periodicity of 180◦.

0	0◦	30◦	60◦	75◦	105◦	120◦	150◦	180◦	210◦	240◦	255◦	285◦	315◦	330◦	300◦	360◦
y = tanѳ	0	0.58	1.73	3.73	-3.73	-1.73	-0.58	0	0.58	1.73	3.73	-3.73	-1	0.58	-1.73	0

Take a scale of 1cm to represent 30◦ on the ѳ- axis and 1cm to represent 1 unit on the y – axis.

Example

Using the same axis, a scale of 1cm to represent 30◦ on the ѳ axis and 2cm to represent 1 unit on the y-axis, draw the graphs of the following relations.

(a) y = sinѳ

(b) y = 2sinѳ

(c)y = ½ sinѳ in the interval 0◦ ≤ ѳ ≤ 360◦.

Solution

(Refer to table 14.3 and Fig. 14.19)

We observe that the curves:

y = sinѳ

y = 2sinѳ

y = ½ sinѳ

have the same periodicity (360◦), but differ in amplitudes.
The amplitude of y = sinѳ is 1.

The amplitude of y = 2sinѳ is 2.

The amplitude of y = ½ sinѳis ½.

In general, the curve y = Asinѳ has amplitude /A/ abd a periodicity of 360◦. This property of the same curve is also a characteristic of the cosine curve.

Evaluation

1. Prove that sec²ѳ + cosec²ѳ = (tanѳ + cotѳ) ².

General Evaluation

(1) Draw the graph of y = 2cosx – 1 in the range 0◦ ≤ x ≤ 360◦ at intervals of 30◦.

(2) Draw the graph of y = 3sin x – 1 in the range of 0◦ ≤ x ≤ 360◦ at intervals of 30◦

(3) Sketch the graph of:

(i) y = sin2x                                                                 (ii) y = cosx

(iii) y = sec x                                                                (iv) cosec x

all at intervals of 30◦ range 0≤ x ≤ 360.