Trig Functions For Any Angle

MathsDirect

When considering right angled triangles, we have only needed to use angles between 0 and 90^o. Trigonometry can however be extended, beyond 90^o. In fact there is no limit on the size of angle that you can apply the trig functions to. You can even find the sin, cos and tan of negative angles. These are all based, however, on the results between 0 and 90^o.

To understand the meaning of trigonometry, beyond this range, it is useful to consider the following diagram.

It is easy to see how we can apply trigonometry to the triangle, where the angle is just the rotation from the x-axis, in an anti-clockwise direction.

This is called the first quadrant,( i.e. the first quarter,) and all the angles range from 0 to 90^o.

If you think in terms of co-ordinates, both o and a are positive.

There is no reason why we cannot continue to move around the circle, in an anticlockwise direction, into the second quadrant.

Our angle is now greater than 90^o, but we can still see a value for o, a, and h, so we still have a way to understand the trig functions.

Notice that, in terms of co-ordinates, a is now negative, while o and h are still positive. As a result, the trig functions that contain the adjacent side,( cos and tan,) will be negative

Before we move on to the next quadrant, let's consider a specific example in the second quadrant.

Find sin, cos and tan, for 120^o.

The angle is 120^o measured anticlockwise from the x-axis. This means that we are dealing with a 60^o triangle. We can work out the trig functions from this.

We just need to bear in mind that cos and tan must be negative, for the reason given above.

We can now move on to the third and fourth quadrants.

Now both a and o are negative. Since h is just a length, we say that it is always positive.

This means that sin and cos are now negative, but tan becomes positive, as the negatives in o and a cancel each other out.

In the last example, we worked with 60^o, (180-120,) In the third quadrant, to find the acute angle to work with, we subtract 180^o from our angle, so for 210^o, we use 30^o. We then just need to make sure that we get the right sign

Now a is positive, but o is still negative. This means that sin and tan are now negative, but cos is positive.

To find the acute angle that we need to work with, subtract your angle from 360^o.

Find sin, cos and tan for 315^o.

Once you go beyond 360^o the whole pattern just repeats itself. For negative angles. You go around the circle in the opposite direction,( i.e fourth quadrant first, then third quadrant.) The same rules still apply, and are summarized below.

Quadrant	Acute Angle to use	Sin	Cos	Tan

You may be asked to find all the solutions to an equation in a given. This is dealt with on the next page.

Go to Next Page

Return to Trig Tutorial Contents