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
- 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.
Weekend Assignment
Given that 4cos x + 3sin x = 5, find the value of
(1) Sin x
(2) Cos x
(3) Tan ѳ
(4) Cot ѳ
(5) Sin x + cos x
Theory
(1) Draw the graph of inverse trig function for sin x –
(2) Find the inverse of the following and their domains
(a) y = sin x (b) y = cos x (c) y = tan x
(d) y = cosec x (e) y = sec x (e) y = cot x
See also