2YTrigQuadrants


 * Trigonometry in 4 Quadrants **[[image:2Yquadnums.gif width="160" height="150" align="right" caption="1. Quadrant Numbers"]]

The cartesian plane is divided into quarters by the x and y axes.

Each quarter is called a ** Quadrant **.

The quadrants are numbered anti-clockwise from the positive x-axis, with the 1st Quadrant being the area where both x and y are positive. (See Fig 1)

From the unit circle definition of the trig functions, x = cos( q) and y = sin( q), it follows that sin will be positive when y is positive and cos will be positive when x is positve.

Since tan( q) is given by the ratio: math . \qquad \tan( \theta ) = \dfrac {\sin (\theta)}{\cos (\theta)} math It follows that tan will be positive when sin and cos both have the same sign (both positive or both negative).

From this, we can draw a simple chart indicating where each trig function is positive. (See Fig 2) This diagram can be remembered using
 * ** CAST ** ...... (or)
 * ** A **ll ** S **tations ** T **o ** C **roydon

Calculating trig ratios

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.


 * 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 (225^\circ) = -\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 (150^\circ) = -\dfrac {\sqrt{3}}{2} math

Angles Larger than 360º.

If the angle is larger than 360º (or 2 **p** ), remember that each 360º (or 2 **p** ) is one complete rotation. Use this to decide which quadrant the final angle is in and proceed as described above. {some people prefer to subtract multiples of 360º to get the angle iin a range they are more familiar with}

Negative Angles

Recall that a negative angle starts from the positive x-axis and rotates clockwise.



Example

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