Pie Graph or Pie Chart

To find the angle of each sector

Total of data corresponds to 360^o.

Let x^o = the angle at the centre for item A, then

The data given in example 1 can be used to draw a pie graph.

Calculation of Angles

Food:

Angle at centre

= 150^o

Rent:

Angle at centre

= 40^o

Similarly we can calculate the remaining angles, and the total of angles column should always come to 360^o.

A survey was conducted to find out the consumption of various brands of soap. The results of the survey are given below:

(i) Soap Cake:

(ii) Soap Powder:

(i) Using the information given in the table for soap cake, draw a pie chart.

(ii) Using the angles in the pie chart for soap powder, complete the table that follows:

(i) Draw Pie Chart

Total of the items = 100

A. Angle at centre

= 216^o

B. Angle at centre

= 72^o

C. Angle at centre

= 54^o

Other Angle at centre

= 18^o

(ii) Complete the table

For P,

Similarly

Same way we can calculate the values of R, S and others.

R = 25%, S = Others =

Graph of y = cosx

Graph:

From the graph it is clear that the curve repeats itself every 360^o (2prad). This fact is expressed by the statement that the function has a period of 360^o (2p radians). In symbols we write cos(x + 360^o.n) = cos(x + 2pn) = cos x where n is a positive or negative integer.

From the graph we also observe that cosx does not pass through the origin. The maximum and minimum values of cosx are +1 and -1 respectively. As x increases from 0^o to 90^o cosx decreases from 1 to 0, as x increases from 90^o to 180^o cosx decreases from 0 to -1, as x increases from 180^o to 270^ocosx increases from -1 to 0, as x increases from 270^o to 360^o cosx increases from 0 to 1. Cosx is period and has a period 2p.

Graph of y = sin x

Now let us construct the graph of y = sinx from x = 0^o to 360^o. The following table is readily constructed for intervals of 30^o.

Plotting the points and drawing a smooth curve through them we have the curve as shown in figure.

Table:

Graph:

From the figure it is evident that the curve repeats itself every 360^o or 2p. This fact is expressed by saying that the function has a period of 360^o or 2p.

In symbols we write sin (x + n.360^o) or sin (x + 2np), sinx = sin (x + n.360^o) = sin (x + 2np), where n is any positive or negative integer. This infers that sinx varies and takes a complete ordered range of values once and that sinx is periodic has the period 2p. From the figure we observe that as x increases from 0^o to 90^o, sinx increases from 0 to 1 and as x increases from 90^o to 180^o, sinx decreases from 1 to 0.

[A function f(x) is periodic with period T if f(x+T) = f(x) for all values of x]

As x increases from 180^o to 270^o, sinx decreases from 0 to -1 and as x increases from 270^o to 360^o, sinx increases from -1 to 0. The maximum absolute value of sin x = 1.

Angles

Fig.(i) Fig.(ii) Fig.(iii)

Let OA and OB two half lines with common end point O. The half lines OA and OB are the sides of an angle and the point O is the vertex of the angle. An angle is an amount of rotation of a half-line (or ray) in a plane about its end point from an initial position to a terminal position.

Measurement of angle

The amount of rotation from initial side to terminal is called the measure of an angle.

Positive and Negative angles

Angles that are formed by counter clockwise (anti clockwise) rotation, such as the one shown in fig (ii) are said to be positive or to have positive measure.

Angles that are formed by a clockwise rotation, like the one in fig(iii) are said to be negative or to have negative measure.

Lines at right angles

The lines are said to be at right angles if the rotating half line (or ray) from starting from initial position to the final position describes one quarter of a circle.

Quadrants

Let X'OX and Y'OY perpendicular coplanar lines intersecting each other at O. We refer X'OX as x-axis and Y'OY as y-axis. It is clear from the adjoining figure, that these two lines divide the plane into four equal parts, each part is called a Quadrant.

The four Quadrants are:

XOY - first Quadrant

YOX' - second Quadrant

X'OY' - third Quadrant

Y'OX - fourth Quadrant

Angle in standard position

If an angle of any measure be given, one can always construct (or draw) a cartesian co-ordinate reference frame in such a way that the origin is at the vertex of the angle, and positive half of the x axis coincides with the initial side of the angle. When this has been achieved, the angle is said to be in standard position. An angle in standard position is said to be in the Quadrant in which its terminal side lies.

vi) An angle is called Quadrant angle if it is in standard position and its terminal side coincides with one of the co-ordinate axis.

Trignometric Ratios

Classification of Triangles According to Sides

Isosceles Triangle

Isoceles triangle is a triangle with following properties and as shown in the figure.

• The two sides of the isosceless triangles are equal.

• The angles opposite to the equal sides are equal.

In the above triangle, sides 'a' are equal and angle 'x' are also equal. Hence it is an Isoceles triangle.

Equilateral Triangle

Equilateral triangle is a triangle with following properties and as shown in the below geometrical figure.

• All the sides of triangle will be equal.

• The angles of triangle will be equal and 60^o

In the above triangle, all the sides of triangle denoted by 'a' are equal and angles are equal to 60^o. Hence the above geometrical figure represents a equilateral triangle.

Scalene Triangle

Scalene triangle is a triangle with following properties and as shown in the below geometrical figure.

• All the sides of triangle will be unequal.
• The angles of triangle will be unequal.

In the above triangle, all the sides of triangle denoted by a, b and c are unequal and angles x, y and z are also unequal. Hence the above geometrical figure represents a scalene triangle.

Median-Altitude

Median

In a triangle, a line joining the midpoint of a side to the opposite vertex is called a median.

AD is a median of ABC.

In any triangle, it can be proved that all the three medians meet at a point. The point where the three medians meet is called the Centroid of the triangle. The point G is the centroid in ABC in the following figure.

Altitude

In a triangle, the perpendicular from a vertex to the opposite side is called the Altitude.

In ABC of the following figure, AD is the altitude.

Three altitudes always meet at a point called Orthocentre of the triangle.

In Fig.(i): AD, BE, CF are the altitudes. O is the orthocentre. O lies inside the acute angled triangle.

In Fig.(ii): AB, CB and BD are the altitudes. B is the orthocentre.

In Fig.(iii): Three altitudes AD, BE, CF are produced to meet at O. O lies outside the obtuse angled triangle.

In a triangle, the bisectors of the three angles meet at a point called the Incentre.

In ABC in the figure above, the bisectors of the three angles meet at I. I is the Incentre.

With I as centre and IM as radius, a circle drawn to touch the sides, is called the Incircle.

In a triangle, the perpendicular bisectors of the three sides meet at a point called the Circumcentre.

In ABC, OD, OE, OF are the perpendicular bisectors of sides BC, AC and AB. O is called the circumcentre.

With O as centre and OA as radius a circle drawn will pass through B and C. Such a circle is called the Circumcircle.

Classification of Triangles According to Angles

(i) If one angle of a triangle is a right angle, then it is called a right angled triangle. Note that the other two angles are acute.

(ii) If all the angles of a triangle are acute, then it is called an acute angled triangle.

(iii) If one angle of a triangle is obtuse, then it is called an obtuse angled triangle. Note that the other two angles are acute.