Mathematics Form 4 Coursework e-Content CD

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Example
Draw a graph of y = cosx0 for -180O n x n180O

Solution
Make a table of xO and cosxO value to be plotted on x and y-axes respectively.


Plot the points on a Cartesian plane and join them with a smooth curve as shown below.

OBJECTIVES
By the end of the lesson you should be able to;
1. Derive trigonometric identity of sin2x + cos2x = 1
2. Draw graphs of trigonometric ratios of the form y=a sinx and y = a cos (bx+0)
3. Solve simple trigonometric equations graphically.
4. Deduce from the graph amplitude, period, wavelength and phase angles.

TRIGONOMETRY (3)


In life, we find various changes that create waves. A wave is a pattern or variation which moves or spread through a medium. Examples of waves include sound waves, light waves and water waves.

There are various types of equations. These include linear quadratic and cubic equations among others. There are equations that connect variables which involves trigonometric values.
E.g. y = sinx
O
, y=2cosxO
and so on.
These equations can be used to draw trigonometric graphs. To draw a trigonometric graph, it is important to make a table of values of the angles and their corresponding trigonometric ratios.
In this topic we will learn how to draw and use trigonometric graphs, to solve trigonometric equations.



Graphs of trigonometric ratios

There are various forms of trigonometric equations that emanate from original equations such as; y = sinx, y = cosx and y = tan x
Examples include y = a sinxO, y=a cos xO and y = a cos (bxO + F).

In this section, we will draw the graphs of the functions;
Y = a sinxo
Y = a cosbxO
Y=a cos (bxO+ 0)

Then use them, to deduce the amplitude, period, wavelength and phase angle.


Example 1


Draw the graph of y = 2sin xo for -180o n xo n180o and deduce the amplitude, period and wavelength.


Solution
Table of values of y = 2 sin xo


Note that the graph has the shape of a wave. It can be referred to as a sign wave.


The amplitude of a trigonometric wave is the maximum displacement or height of the wave above or below the x-axis. In this case the amplitude is 2.

The period of aware is the cycle within which the wave starts to repeat itself. The cycle in this graph occur after 360o
. From point A the wave starts to repeat itself at point B. Therefore the period of the wave is 360o
or 2P
radians.
The wavelength is the distance between two consecutive corresponding points on a wave. It can be obtained easily by finding the distance from one point at the top of a wave to the next top point. In this case the wavelength is 4cm.

Example 2
Draw the graphs of the function. y = 3 cos2xo and y = 3cos (2x + 60o) on the same axes for -180o n xon + 180o and deduce the amplitude, wavelength, period and phase angle.

Solution
Table of values for y = 3cos 2xo

Table of values for y = 3cos (2x + 60o)



Amplitude is 3
Wavelength is 6cm
Period is 180o
Phase angle is -30o ie the wave moves 30o in the negative direction of x -axis.

Note: The equation y = 3cos 2x + 60o can be re-written as y = 3cos 2 (x +30o)
The angle 30oobtained inside the brackets when 2 is factored out.
When the angle inside the brackets is +FO,wave moves FOin the negative direction of x - axis and therefore the phase angle is - FO.

Example 3
Draw the graphs of y=3sinxo and y = 2cos 2xo for 0on xn 360o on the same axes and
(a)Find the values of x in which 3 sin x = 2cos 2x
(b) Solve y = 2 cos2 x for values of x in which 0o n x n180o

Solution
Table of values


(a)The answer are real at the point of intersection of the two curves.


X = 25o +/- 1o and 154o +/-1o


(b) The answers are read at the points of intersection of the curves y = 2cos 2x and line y = 0 (x - axis )
X = 45
o
and 135o
.

Example 3
Draw the graphs of y=3sinx0 and y = 2cos 2x0 for 00≤ x≤ 3600 on the same axes and
(a) Find the values of x in which 3 sin x = 2cos 2x
(b) Solve y = 2 cos x for values of x in which 00 ≤ x ≤1800

Solution
Table of values
X 0 450 900 1350 1800 2250 2700 3150 3600
y= 3sin x 0 2.1 3 2.1 0 -2.1 -3 -2.1 0
Cos 2x 1 0 -1 0 1 0 -1 0 1
y=2cos 2x 2 0 -2 0 2 0 -2 0 2

(a) The answer are real at the point of intersection of the two curves.

X = 250 + 10 and 1540 + 10

(b) The answers are read at the points of intersection of the curves

y = 2cos 2x and line y = 0 (x – axis )
X = 450 and 1350.

Example 1

Draw the graph of y = 2sin x0 for -1800 x0 1800 and deduce the amplitude, period and wavelength.

Solution

Table of values of y = 2 sin x0 X0 -3600 -2700 -1800 -900 00 900 1800 2700 3600 Sin x 0 0 1 0 -1 0 1 0 1 0 Y = 2 sin x0 0 2 0 -2 0 2 0 2 0

Note

The graph has the shape of a wave. It can be referred to as a sign wave. The amplitude of a trigonometric wave is the maximum displacement or height of the wave above or below the x-axis. In this case the amplitude is 2. The period of aware is the cycle within which the wave starts to repeat itself. The cycle in this graph occur after 3600. From point A the wave starts to repeat itself at point B. Therefore the period of the wave is 3600 or 2π radians. The wavelength is the distance between two consecutive corresponding points on a wave. It can be obtained easily by finding the distance from one point at the top of a wave to the next top point. In this case the wavelength is 4cm.


OBJECTIVES
1. By the end of the lesson you should be able to;
2. Derive trigonometric identity of sin2x + cos2x = 1
3. Draw graphs of trigonometric ratios of the form y=a sinx and y = a cos (bx+0)
4. Solve simple trigonometric equations graphically.
5. Deduce from the graph amptitude, period, wavelength and phase angles.


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