# 5 - Trigonometry

Take two perpendicular and graduated lines. Then draw a circle of radius 1 and whose center is the intersection of the lines.

Now, draw a line passing through O ; the angle between this line and the horizontal axis is 40 °. Denote by M the intersection of this line and the circle. The abscissa of M is cos(40) and its ordinate is sin(40).

The angle you chose doesn't matter : M is on the circle, you'll always obtain :

When you move M to the right, its cosine gets closer to 1 and its sine gets closer to 0 ; when it goes up, its cosine gets closer to zero and its sine gets closer to 1. Sine et Cosine are functions : give them angles (degrees or radians), they will give you numbers between -1 and 1 in return.

The radian is a unit of angular measurement. If the length of the red arc is x, then the angle measures x radians.

The whole circle measures 360 degrees. In radians, it measures the circumference of the circle, that is 2 (the radius of the circle is still 1). In the drawing, the angle x is approximately 1 radian. As you know that 2 rad=360°, you can convert, thanks to a cross-product, degrees into radians and radians into degrees. Hence :

## Table of values : sine and cosine

You have to know by heart the cosine and sine values of the angles below (to learn them, you can learn the drawing).

 angle x 0 cos x 1 0 - 1 sin x 0 1 0

 Thanks to Pythagoras' theorem, in the triangle ONM which is right-angled at N, we have : , and since the radius of the circle is 1, we have :

## Graph

If angles are greater than 2, the point M turns around the circle ; when it is 2 radians, it comes back to the same place. Hence :

For example , so you can draw, thanks to the table of values, the graph of the cosine function, and the graph of the sine function :

You can prolong the two graphs infinitely. On the graph, we can see than cosine is even (symmetric with respect to the Y-axis ), and the sine is odd (symmetric with respect to the origin).

## Tangent

We won't say much about the tangent function on this page. Nevertheless, you have to know that :