Angles exceeding 90 degrees

We have defined sin(), cos() and tan() for 0 < < 90.

However, these functions can be defined for any .

We only need to define sin() and cos() for all numbers, because then we know what tan() is because of

Draw a circle of radius 1 about the origin, and mark the point A (x, y) on the circle, such that OA makes an angle of with the x-axis, where the angle is measured anti-clockwise, and 0 < < 90.

Also let B be the point on the x-axis that makes ABO a right angle.

A is the point (x, y), whcih means that OB = x and AB = y.

Now let us see what sin() and cos() are.

          =

          = y

           =

           = x

This means that sin() is the y-coordinate of the point on the circle, and that cos() is the x-coordinate of the point on the circle.

Suppose we want to find sin() or cos() for any .

Draw a circle of radius 1.

Mark the point A on the circle such that OA makes an angle of with the x-axis, measured anti-clockwise.

Then sin() is the y-coordinate of A and cos() is the x-coordinate of A.

This definition coincides with our previous definition of sin() and cos() for 0 < < 90.

And it allows us to define sin() and cos() for any .

Here, however, we shall only be interested in values of for which 0 < < 180.

But, by using this definition, we have defined sin() and cos() for any .

We have even defined it for negative . If is negative then you go backwards degrees, ie. Clockwise.

The unit circle definition tells us what sin(0) and cos(0) are.

Remember that the circle starts at (1, 0) and goes up and to the left.

If is 0 that means that the curve is at the starting point.

So (cos(0), sin(0)) = (1, 0)

ie. cos(0) = 1 and sin(0) = 0.

Similarly, cos(90) = 0 and sin(90) = 1.

One point of difference with our extended definition is that now sin and cos can become negative.

The y-coordinate of the circle is positive for 0 < < 180, so sin() > 0 for 0 < < 180.

But the x-coordinate of the circle is negative for 90 < < 180.

This means that cos() is negative for 90 < < 180.

We can now calculate exact values for the angles 0^o and 90^o.

sin(0) = 0         cos(0) = 1        tan(0) = 0

sin(90) = 1       cos(90) = 0      tan(90) does not exist.

The following formulas are true for any .