22Dunitcircle

toc = The Unit Circle =

This is where the formal definition of sine, cosine and tangent come from.

Draw a circle on a set of axes with its centre at the origin and with a radius of 1 unit.
 * Because the radius is 1 unit, we call this ** The Unit Circle **.

Draw in a line from the origin to the point (x, y) on the circumference of the circle.
 * This is the radius of the circle so the line has a length of 1 unit.

Define q (theta) as the angle from the __**positive x-axis**__, __**anticlockwise**__ to the line.

Quadrants
From the diagram, we can see that the axis lines divide the circle into 4 equal quarters or ** quadrants **.

Given that q is measured anticlockwise from the positive x-axis, we number the quadrants from 1 to 4 in order as q increases.
 * Quadrant 1: .... 0º < q < 90º
 * Quadrant 2: ... 90º < q < 180º
 * Quadrant 3: .. 180º < q < 270º
 * Quadrant 4: .. 270º < q < 360º


 * Notes: **
 * Angles bigger than 360º take us back into Quadrant 1 etc
 * Measuring clockwise down from the positive x-axis creates negative values of q.

Formal Definitions

 * Cosine of q ** **{cos( q )}** is defined as the x-coordinate of the point (x, y) on the circle.
 * Sine of q ** ** {sin( q )} ** is defined as the y-coordinate of the point (x, y) on the circle.

If we take our unit circle and draw in a right-angled triangle with we can see that our trig rules conform to these definitions (because the hypotenuse = 1)
 * the origin (0, 0) and the point (x, y) on the circle as vertices
 * and the x-axis as the base,

math . \qquad \sin \big( \theta \big) = \dfrac{\text{OPP}}{\text{HYP}} = \dfrac{\sin \big( \theta \big) }{1} math

math . \qquad \cos \big( \theta \big) = \dfrac{\text{ADJ}}{\text{HYP}} = \dfrac{\cos \big( \theta \big) }{1} math


 * Note **
 * From the definition of q, we see that sin and cos are not restricted to angles between 0º and 90º.

4 Quadrants
Keep in mind that
 * cos( q ) = x-coordinate
 * sin( q ) = y-coordinate

Because the x and y coordinates can be either positive or negative, we can see that sin and cos can be either positive or negative.

You calculator knows this. Eg, typing in **cos(110)** gives

math . \qquad \cos \big( 110^\circ \big) = -0.3420 math

{110º is in the 2 nd quadrant where cos is negative}

Tangent
Using the right angled triangle in the diagram to the right, and using the trig rule for tan, we get that:

math . \qquad \tan \big( \theta \big) = \dfrac { \text{OPP}} {\text{ADJ}} = \dfrac {\sin \big( \theta \big) }{ \cos \big( \theta \big) } math

This is an __**identity**__,
 * meaning that it is true for all values of q.

math . \qquad \tan \big( \theta \big) = \dfrac {\sin \big( \theta \big) }{ \cos \big( \theta \big) } math

Recall that
 * When multiplying or __**dividing**__:
 * if the signs are **the same**, the answer is **positive**
 * if the signs are **different**, the answer is **negative**

Therefore, we can conclude that in the four quadrants tan will either be positive or negative depending on the values of sin and cos:

Trig Functions in 4 Quadrants
We can simplify the diagram to the right by only listing the functions which are __**positive**__ in each quadrant.



This diagram can be remembered using
 * ** CAST ** ...... (or)
 * ** A **ll ** S **tations ** T **o ** C **roydon

To calculate trig ratios in all four quadrants, follow these four steps.
 * 1) Draw a simple diagram of the angle, anticlockwise from the positive x-axis.
 * 2) Find the angle to the __closest__ part of the x-axis.
 * 3) Calculate the value of the trig ratio for the angle found in step 2 (use exact value if possible).
 * 4) Use ** CAST ** to establish whether the trig ratio should be positive or negative.

Recall that the exact values are:


 * Example 1a **

Find the exact value of sin(225º)


 * 1) Draw diagram
 * 2) Angle to closest x-axis is 45º
 * 3) sin 45º is 1/sqrt(2) (from exact values)
 * 4) In 3rd quadrant, so sine is negative.

Thus answer is: math . \qquad \sin \big( 225^\circ \big) = - \dfrac{1}{\sqrt{2}} math


 * Example 1b **

Find the exact value of cos(150º)


 * 1) Draw diagram
 * 2) Angle to closest x-axis is 30º
 * 3) cos(30º) is sqrt(3)/2 (from exact values)
 * 4) In 2nd quadrant, so cos is negative.

Thus answer is: math . \qquad \cos \big( 150^\circ \big) = - \dfrac{ \sqrt{3}}{2} math .