**Graphs of Trigonometric Function**

The graph of the followings will be considered

Table of Contents

(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.

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**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.

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**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.

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**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◦.

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**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 = *A*sinѳ has amplitude /*A*/ abd a periodicity of 360◦. This property of the same curve is also a characteristic of the cosine curve.

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**Evaluation**

- Prove that sec
^{2}ѳ + cosec^{2}ѳ = (tanѳ + cotѳ)^{ 2}.

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**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.

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**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

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**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

Cubic equations and their factorization