# Trigonometrical Ratios

Describes the derivation of the basic ratios and their relationship one with another

**Contents**

## Trigonometric Ratios For Any Angle.

A knowledge of the Pythagoras theorem and the properties of Similar triangles, is assumed.## Definition Of The Basic Ratios.

Consider the two right angled triangles shown. As they are equiangular the following relationships exist between the lengths of their sides. Diagram
Re-arranging:-
These two ratios are clearly independent of the size of the triangle and depend solely upon the size of the angles of a right angled triangle. By definition the value of equalities shown in equation (2) is called .
Using the same analysis two other ratios can be identified. These are
In Conclusion, considering the following right angled triangle:-

## The Relationships Between Sin; Cos And Tan.

By Inspection of equations (3:4:5)it can be seen that In addition if we put a = 1 . The following diagram can be drawn. The two perpendicular Coordinates are Ox and Oy. The radius OP is of unit length and the angle xOP is measured clockwise from xOP and the coordinates of P are defined as whatever the position of P.## Special Angles

From the definitions and from an inspection of the graph it is possible to rite down the Ratios for the following angles.NOTE * The prefix co- in the ratios stands for "complementary" and means that the ratio of any angle is equal to the co-ratio of the complementary angle. e.g.*

## Angles Larger Than 90 Degrees.

From the definitions and from the graph it can be seen that:-- In the first quadrant, sin is +ve; cos is +ve; tan is +ve.
- In the second quadrant, sin is +ve; cos is -ve; tan is -ve.
- In the third quadrant, sin is -ve; cos is -ve; tan is +ve.
- In the forth quadrant, sin is -ve; cos is +ve; tan is -ve.

## The Graphs Of The Trigonometrical Ratios

In defining the ratios the following graph was used. It can be seen that the value of is given by the y-ordinate. Thus by drawing a circle of unit radius, the value of of the sine of any angle can be found. In this way the following graph was drawn. On the left hand circle is measured from Ox in an anticlockwise direction whilst on the right hand graph is measured along the x axis in the normal way. the values taken by the cosine as the angle increases from 0 to 90 degrees will be the same as those taken by the sine as the angle decreases from 90 degrees to 0. The two graphs are identical in shape and magnitude but displaced by 90 degrees.To construct the graph of is slightly more complicated. By inspection it can be seen that as:-

The tan of the angle will be infinite whenever the cosine is zero. To construct the graph, CX is drawn at unit length. The points are markedoff on the line XY and correspond to the various angles chosen for CP. If a point Q is plotted such that its abscissa on ON is equal to the number of degrees in the chosen angle XCP. This process is repeated for each value of the angle XCP.