Trigonometry with any angle

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Trigonometry with any angle

There are three basic trigonometric functions for acute angles: Sine (Sin), Cosine (Cos), and Tangent (Tan).

When using a right-angled triangle we get:

Trigonometry with any angle

These functions have a unique value for an acute angle that can be obtained from a scientific calculator.

These formulae are only applicable for an acute angle in a right-angled triangle, and so the next stage is to extend to work with any angle in radians and degrees.

On a coordinate grid a general angle is measured from the positive x-axis and is represented by the angle through which a line OM rotates about the origin.

When we rotate anti-clockwise, the angle is positive while a clockwise rotation gives a negative angle.

Trigonometry with any angle

The four quadrants of the Cartesian axes are as follows:

Trigonometry with any angle

As the line OM rotates, the point M moves to the first quadrant where its coordinates are both positive, and into the second quadrant, where the x-coordinate becomes negative.

In the third quadrant, both coordinates are negative and finally, in the fourth quadrant, the point has a positive x- and negative y-coordinate. (See below.)

Measure of central tendency

You can see that the angle MON, called a, is always acute, and measured from the x-axis.

For example:

The 4 quadrants

The signs of the trigonometric functions depend on which quadrant the point M lies in and represent the signs of the x- and y-coordinates of M.

Learn the information in the following diagrams to help you understand the signs.

First quadrant

All the functions are positive.

1st quadrant

Second quadrant

2nd quadrant

By looking at the signs of the coordinates of M, we see that the trigonometric functions of are:

2nd quadrant

Third quadrant

3rd quadrant

The signs of the coordinates of M show us that the trigonometric functions are:

3rd quadrant

Fourth quadrant

4th quadrant

The signs of the coordinates of M show us that the trigonometric functions of are:

4th quadrant

This can be summarised as:

Trigonometry with any angle

These sign rules and the value of the acute angle a allow you to find the value of any trigonometric function of any angle.


Find the values of sin 150, sin 210 and sin 690 if sin 30 = 0.5.

sin 150 = sin 30 = 0.5

sin 210 = - sin 30 = - 0.5

sin 690 = sin 330 = - sin 30 = -0.5

You also need to be aware of negative angles created from the rotation of M in a clockwise direction.


i.e. each position of line OM gives us two different values of theta, one that is positive and one that is negative.



Here a = 200 so both angles have the same trigonometric functions.


sin 1600 = sin (-2000 ) = + sin 200

cos 1600 = cos (-2000 ) = - cos 200

tan 1600 = tan (-2000 ) = - tan 200