EUCLID'S AXIOMS AND POSTULATES

AXIOMS

1. Things which are equal to same thing are equal to one another.
2. If equals are added to equals, the wholes are equal.
3. If equals are subtracted from equals, the remainders are equal.
4. Things which coincide with one another are equal to one another.
5. The whole is greater than a part.
6. Things which are double of the same things are equal to one another.
7. Things which are halves of the same things are equal to one another.

POSTULATES

1. A straight line may be drawn from any one point to any other point.
2. A terminated line can be produced infinitely.
3. A circle can be drawn with any centre and any radius.
4. All right angles are equal to one another.
5. If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then two straight lines, if produced infinitely, meet on that side on which the sum of angles is less than two right angles.

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WHAT IS ADDITIVE INVERSE?

Definition

The additive inverse of a number is what you add to a number to create the sum of zero. So in other words, the additive inverse of x is another number, y, as long as the sum of x + y equals zero. The additive inverse of x is equal and opposite in sign to it (so, y = -x or vice versa). For example, the additive inverse of the positive number 5 is -5. That's because their sum, or 5 + (-5) = 0.

What about the additive inverse of a negative number? Using the same approach, if x is a negative number, then its additive inverse is equal and opposite in sign to it. This means that the additive inverse of a negative number is positive. For instance, if x equals -12, then its additive inverse is y = 12. We can verify that the sum of x + y equals zero, since when x = -12 and y = 12, we have -12 + 12 = 0.

It should be noted that the additive inverse of 0 is 0. Zero is the only real number, which is equal to its own additive inverse. It is also the only number for which the equation x = -x is true.

Graphical Representation

We can also think of the additive inverse visually. Let's consider the real number line, which is usually drawn horizontally, with 0 near the middle, the negative numbers to its left, and the positive numbers on the right. Two numbers of opposite sign fall on either side of 0 on the number line at equal distance.

Once we place the point corresponding to a number x on the number line, we know that the additive inverse, or -x, will fall on the opposite side of the number line with respect to 0. In fact, the point 0 is the midpoint between x and its additive inverse -x. For example, when x = 5, its additive inverse is -5.

A number and its additive inverse are equidistant from the 0

It is clear to see that the point 0 is the midpoint of the segment between -5 and 5.

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WHAT IS MULTIPLICATIVE INVERSE?

Multiplicative Inverse is a another word for Reciprocal.

What you multiply by a number to get 1 

Example: 8 × (1/8) = 1

In other words: when we multiply a number by its "Multiplicative Inverse" we get 1.

But not when the number is 0 because 1/0 is undefined!



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WHAT IS TRIGONOMETRY?

INTRODUCTION

In this chapter, we intend to study an important  branch of mathematics called '' TRIGONOMETRY''.

The word TRIGONOMETRY is derived from the Greek words: (i) trigonon and, (ii) metron. The word trigonon means a triangle and metron means measure. Hence, trigonometry means the science of measuring triangles.

TRIGONOMETRIC RATIOS AND ITS FORMULAS

  Ratios           Formulas

1. sinq = Perpendicular/Hypotenuse

2. cosineq = Base/Hypotenuse

3. tangentq = Perpendicular/Base                                                              

4. cosecantq = Hypotenuse/Perpendicular

5. secantq = Hypotenuse/Base                                               

6. cotangentq = Base/Perpendicular

 The perpendicular side is that the side which is front of the angle that we talk about.The hypotenuse side is the side front of right angle. and the remaining side is the base.

IDENTITIES

sin(θ) = 1/csc(θ)

cos(θ) = 1/sec(θ)

tan(θ) = 1/cot(θ)

csc(θ) = 1/sin(θ)

sec(θ) = 1/cos(θ)

cot(θ) = 1/tan(θ)

cot(θ) = cos(θ)/sin(θ)

                                       sin² θ + cos² θ = 1                              

                                              sin² θ = 1 − cos² θ                                   
                                              cos² θ = 1 − sin² θ                                   
                                              tan² θ + 1 = sec² θ                                   
                                              tan² θ = sec² θ − 1                                   
                                              cot² θ + 1 = csc² θ                                   
                                              cot² θ = csc² θ − 1                                   

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