Scientific Notation Rules

 Often we come across abnormally large numbers or abnormally small numbers like 400,000,000,000 or very, very small number like0.00000000002. Especially in the field of science such abnormal size of numbers is quite common.

In calculations involving such numbers, the work is cumbersome and possible errors may occur.

A special type of notation is used to simplify the dealings with these type of numbers and it is named as scientific notation as such numbers are very predominant in the field of science.

However a scientific notation must be used following a set of rules, universally accepted.

Let us take a closer look.
Scientific Notation Rules – Description

Since very large or very small numbers can be rounded to the appropriate place value, the power of 10 is used as base in scientific notation. The established general form is,

                                       a x 10n, where,  1= I a I < 10

If  ‘n’ is positive, it represents the number of zeroes from ‘a’ to the right of unit place and hence  it denotes large numbers.

For very, very small decimal numbers ‘n’ is negative and represents the number of zeroes before ‘a’ up to the decimal point.
Scientific Notation Rules – Examples

The velocity of light is about 310,000,000 kilometers per second.

We find there are seven 0s after the digits 1. That is the number is same as 31 times 10 to the power 7. But as per scientific notation rules, 31x 107 is not allowed and hence the given is modified as 3.1 x 108

Thus in scientific notation, 310,000,000 = 3.1 x 108

Similarly it is equally incorrect to mention (as a scientific notation) 310,000,000 as 0..31 x 109

Let us consider another example with a very  small number.

The wave lengths of a certain color is experimentally found to be 0.0000015 millimeter.

There are 6 decimal places to cross the digit 1from right. That the given number is (1/1000000)th of 1.5.

Hence in scientific notation it is written as 1.5 x 10-6

Importance of Derivatives of Trigonometric Functions

Trigonometric Functions 
In Mathematics, trigonometric functions or circular functions are ratios of the sides of a right angled triangle, having the given angle.  Trigonometry deals with functions of an angle present in a triangle. It relates the angles of the given triangle with the length of its three sides. It also helps to find the length and angle of a triangle. Trigonometric functions are widely used in real life in various fields such as carpentry, surveys, engineering, physics, navigation, astronomy etc.

Trigonometric functions are sin p, cos p, tan p, cot p, sec p, csc p.  Among them sine, cosine and tangent are the most familiar and basic trigonometric functions. Before going into derivatives of trigonometric functions, let us first see what these trigonometric functions refer to.

Let us consider a circle of radius 1. If a ray originates at the origin which makes an angle with x axis, what will the sine, cosine and tangent of this angle provide?
• The sine of this angle gives the length to which the triangle has risen on the other side i.e. the y component

• The cosine of this angle gives the x component length

• The tangent of this angle gives the slope of the triangle, which is obtained by dividing Y component by the X component.

Thus, trigonometric functions can also be defined as the lengths of all line segments from a given unit circle.

Derivative of Trigonometric Function
With this basic understanding on trigonometric functions, let us explore on the derivatives of trigonometric functions namely sin p, cos p, tan p, cot p, sec p, csc p.

The derivatives for each of these trigonometric functions can be represented as follows:

Derivative of Sinx
The symbolic expression of Sinx derivative is given by

d/dx (sin x) = cos x   or sin’ x = cos x


It is easy to find the derivatives of sin x. This is because, the derivative of sin x can be obtained by just calculating the cos x.

Derivative of Cosx
The Cosx derivative is given by

d/dx (cos x) = - sin x or cos’ x = - sin x


Derivative of Tan p
The derivative of tan p is given by “tan-1 p = sec^2   p”

Derivative of Cot p
The derivative of cot p is given by “cot-1 p = - coesc^2 p”

Derivative of Sec p
The derivative of sec p is given by “sec’ p = sec p tan p”

Derivative of Csc p
The derivative of csc p is given by “csc’ p = - csc p cot p”