# 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:**

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.

**The four quadrants of the Cartesian axes are as follows:**

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.)**

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

**For example:**

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.**

**Second quadrant**

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

**Third quadrant**

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

**Fourth quadrant**

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

**This can be summarised as:**

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

**Example:**

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.

**Example:**

Here a = 20^{0} so both angles have the same trigonometric functions.

**Therefore:**

sin 160^{0} = sin (-200^{0 }) = + sin 20^{0}

cos 160^{0} = cos (-200^{0} ) = - cos 20^{0}

tan 160^{0} = tan (-200^{0} ) = - tan 20^{0}