maths portion for 9th std cbse

• Irrational Numbers
• Polynomials
• Ratio and Proportions
• Linear Equation in Two Variables
• Percentage and its applications
• Compound Interest
• Banking
• Lines, angles and triangles
• Congruence of triangles
• Inequalities in a triangle
• Parallelograms
• Loci and concurrent lines in a triangle
• Areas
• Geometrical Constructions
• Trigonometry
• Mensuration of plane figures
• Mensuration of plane figures
• Mensuration of solid figures
• Statistics

Geometrical Constructions

Introduction Making accurate drawings is not a skill confined to artists. Draftsmen, architects, engineers, and designers need to be able to construct geometrical figures.

In this unit we will look at how to carry out some common geometric constructions using only a pair of compasses and a ruler.

Equidistance

In the map shown in Fig.1 below, the town of Richley is 5 km from both Bridgeford and from Adleborough. We say that Richley is equidistant from Bridgeford and from Adleborough.

Figure 1. Three towns marked on a map.

When we are told only that a place is say 8 km away from us we do not know in what direction it lies, so we do not know its exact position. However, from this information we can find all the possible positions of the place. If we plot a

point

• A point has no properties except position. It is an object with zero dimensions.
• Points in the x-y plane can be specified using x and y coordinates.

point the same distance away in every possible direction, we find we have drawn a circle. Click somewhere in the animation space below to set a distance from the starting point and see what happens.

Figure 2. The possible positions of a point a given distance away.

If you were told that Ceyton was

equidistant

Items that are at an equal distance from an identified point, line or plane are said to be equidistant from it.

equidistant from Ayton and Beeville, and you were given the following map, can you see where the possible positions of Ceyton would be? The circles represent all the points a given distance from Ayton and all the points the same distance away from Beeville. Try dragging the green handle to alter the size of the red circles (they should always be the same size as one another).

Figure 3. Map of the Ayton area.

Trigonometry

The study of angles and the angular relationships of planar and three-dimensional figures is known as trigonometry. The trigonometric functions (also called the circular functions) comprising trigonometry are the cosecant , cosine , cotangent , secant , sine , and tangent . The inverses of these functions are denoted , , , , , and . Note that the notation here means inverse function, not to the power.

The trigonometric functions are most simply defined using the unit circle. Let be an angle measured counterclockwise from the x-axis along an arc of the circle. Then is the horizontal coordinate of the arc endpoint, and is the vertical component. The ratio is defined as . As a result of this definition, the trigonometric functions are periodic with period , so

(1)

where is an integer and func is a trigonometric function.

A right triangle has three sides, which can be uniquely identified as the hypotenuse, adjacent to a given angle , or opposite . A helpful mnemonic for remembering the definitions of the trigonometric functions is then given by "oh, ah, o-a," "Soh, Cah, Toa," or "SOHCAHTOA", i.e., sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, and tangent equals opposite over adjacent,

			(2)
			(3)
			(4)

Another mnemonic probably more common in Great Britain than the United States is "Tommy On A Ship Of His Caught A Herring."

From the Pythagorean theorem,

(5)

It is therefore also true that

(6)

and

(7)

The trigonometric functions can be defined algebraically in terms of complex exponentials (i.e., using the Euler formula) as

			(8)
			(9)
			(10)
			(11)
			(12)
			(13)
			(14)
			(15)
			(16)
			(17)
			(18)

Hybrid trigonometric product/sum formulas are

			(19)
			(20)
			(21)
			(22)

Osborne's rule gives a prescription for converting trigonometric identities to analogous identities for hyperbolic functions.

For imaginary arguments,

			(23)
			(24)

For complex arguments,

			(25)
			(26)

For the absolute square of complex arguments ,

			(27)
			(28)

The complex modulus also satisfies the curious identity

(29)

The only functions satisfying identities of this form,

(30)

are , , and (Robinson 1957).