The Right Angle Triangle

CHAPTER ZERO

SOME BASIC MATHEMATICS

TRIGONOMETRY

The Right angle Triangle

From Pythagoras Theorem,

In addition, ,

, and

However,

Therefore, 0.1

Also,

Using Pythagoras Theorem, we conclude that

0.2

Though we have proved equations 0.1 and 0.2 for a right-angle triangle, they are indeed true for any angle .

Next, we draw the graphs of the trigonometric functions:

Fig. 1.1: The graphs of sine, cosine and tangent of ,

If the y-axis of the sine function is shifted forward by 900, we shall obtain the graph for cosine function. We can therefore say that

. 0.3

Similarly, we infer that

0.4

From equation 0.3, we note that is ahead of , or in other words, lags by 900.

We recall that

Thus,

Again, we arrive at the result in equation 0.3. You can easily verify that equation 0.4 also holds true.

Note that

In other words, the Sine and Cosine functions repeat after every 3600, while the Tangent function repeats itself after every 1800. We therefore say that Cosine and Sine functions have a period of 3600, while the Tangent function has a period of 1800.

From which it readily follows that

As an example,

0.5

You can easily show that

from which it follows immediately that

0.6

From equations 0.5 and 0.6, we conclude that

EXPONENTIAL FUNCTIONS

The series expansion of the exponential function ex is

0.7

To get the expansion for e-x, replace x by –x in equation 0.7

0.8

Adding equations 0.7 and 0.8 and dividing the result by 2 gives

0.9

But the right hand side of equation 0.9 is the series expansion of .

Hence,

Equation 0.8 –equation 0.9 yields

The square root of the number –4 is

, where . is an example of an imaginary number.

;

Higher powers of just repeat this set of four numbers. Hence, we can write a sequence

1, ….

We then notice that this sequence looks like that of , or , taken in steps 900 from 00 to 3600. Respectively, these are:

0, 1, 0, -1, 0, ………, and

1, 0, -1, 0, 1, ……….

Thus, we expect that there might be a relationship between the number , and and . This is indeed so.

Replacing by in the expansion for , we get

0.10

Similarly,

0.11

From equations 0.10 and 0.11, we obtain

0.12

and

0.13

The right hand sides of equations 0.12 and 0.13 are, respectively, the series expansion for and.

Therefore,

Similarly,

DIFFERENTIATION

(i) Polynomials

Given the function , where a is a constant, the differential of with respect to is . We write

Examples: 1. Find the differential of with respect to .

2. Differentiate with respect to t.

(2) Exponential and Trigonometric functions

We can differentiate the expansion for an exponential function to get the differential of the function. For instance,

Let

Then, differentiating term by term, we get

Indeed, the Maclaurin series for is

(where )

It should now be easy to differentiate trigonometric functions.

, since

It is left to the student to show that

INTEGRATION

(1) Polynomials

We have seen that , where a and n are constants. Multiplying both sides by gives

Integrating both sides,

where c is an arbitrary constant, called the constant of integration.

Thus,

That is,

, n

This can also be written as , n. Notice that we still put c, and not –c. This is because c is arbitrary.

Example: Integrate with respect to x.

(2) Exponential and Trigonometric functions

Recall that

Then,

Therefore,

Example: Integrate with respect to x.

which means that

Trigonometric functions, being sums of exponential functions are easy to integrate as all we need to do is to integrate the terms in the sum one by one.

Example: Integrate with respect to t.

+ c

+ c+ c