Trigonometric functions

Trigonometric functions

We shall only consider the four basic trigonometric functions

Sin and Cos

the independent variable x being measured in radians, i.e. taking as unit angle the angle subtended at the center of a circle by a segment equal in length to the radius.

The graph of the function y = sin x is shown in Fig. 34. It is evident from the formula:

that the graph of y = cos x (Fig. 35) can be obtained from the graph of y = sin x simply by displacing it to the left along the axis X by an amount  

Tan

The graph of the function y = tan x is shown in Fig. 36. The curve consists of a series of identical, separate, infinite branches. Each branch is located in a vertical strip of width  and consists of an increasing function of x. Finally, Fig. 37 gives the graph of y = cot x, which is also made up of separate infinite branches.

Properties of trigonometric functions

The graphs obtained by displacing the graphs of y = sin x and y = cos x to the left or right along the axis X by an amount 2 , coincide with the original graphs, corresponding to the fact that functions sin x and cos x have period 2 , i.e.

for any x. The graphs of functions y = tan x and y = cot x similarly coincide on displacement along the axis X by the amount .

The graphs of the functions:

are always similar to those of y = sin x and y = cos x. For example, to obtain the graph of the first of functions (17) from the graph of y = sin x, the lengths of all the ordinates of this latter function must be multiplied by A, and the scale on the axis X changed in such  a way that the point with abscissa x becomes the point with abscissa x/a. Functions (17) are also periodic, with period 2 /a. 

The graphs of the more complicated functions:

which are referred to as simple harmonic curves, are obtained from the graphs of functions (17) by displacement to the left along the axis X by an amount b/a (taking b > 0). Functions (18) also have a period of 2 /a.

The graph of another more complicated function:

consists of the sum of several terms of type (17), and can be constructed, for example, by adding the ordinates of the graphs of the separate terms. The curves thus obtained are usually referred to as compound harmonic curves. Figure 38 illustrates the construction of the graph of the function:

It may be remarked here, that the function:

can be written in the form (18) and represents a simple harmonic oscillation.

We write, in fact:

so that, from trigonometry, an angle b1 can always be found such that:

Substituting for A1 and B1 in (19) from expressions (20), and using (21), we obtain:

the source:

A COURSE OF Higher Mathematics VOLUME I. SMIRNOV.

By: Fady tarek