The CAST Diagram

Angles can be measured from 0 degrees on a set of x-y axes, in a counterclockwise direction. Angles in standard position like this have trigonometric fractions associated with them, in the following way:
  • If the angle is less than 90 degrees, the trig fractions for the angle are made in the ordinary way.
  • If the angle is greater than 90 degrees, you use the trig fractions for the reference angle .
The 'reference angle' is the angle from to the nearest x axis.
is always positive.
The values of sin, cos, and tan for will be the same as the values for
These can be worked out, according to the rule:

However, depending on which quadrant is in, these trig values will be made using sides which may be negative.
On this page, we will show you how this happens, and then give you a 'shortcut' for remembering which trig functions are negative in which quadrant!

Just as a reminder, here is how the quadrants are numbered:

Now here's an angle in the first quadrant:
All three sides of the triangle are positive. This means that sin, cos, and tan are all positive.

Now let's look at an angle in the second quadrant:

Because the triangle is drawn to the nearest x axis, it is drawn to the left. This means that the adjacent side is negative.
So when the three trig fractions are calculated, sin is positive.
cos and tan will be negative.
(In case you were wondering, the hypotenuse is always considered to be positive. If you were to work out its length, using the Pythagorean Theorem, the negatives on the other two sides would disappear as soon as you squared them!)

Now on to the third quadrant:

This time the triangle is drawn left and down. This makes both the adjacent side and the opposite side negative.
As a result, all three trig fractions will involve negative numbers:
sin and cos will be negative.
Tan, however, is positive!

Finally, what happens in the fourth quadrant:

Because of where the triangle is this time, it is clear that only the opposite side will be negative.
This means that sin will be negative, as well as tan.
Cos is positive.

A little confused? Well, at this point you should be aware of the reasons why sin, cos, and tan are sometimes positive, and sometimes negative, depending on which quadrant angle is in. Any trig fractions using one negative side will be negative!
But how can we remember all of this? That's the easy part! There is a short acronym and diagram that will help you to remember which trig function is positive in which quadrant.

Let's summarize first:
  • Quadrant 1: All were positive
  • Quadrant 2: Sin was positive
  • Quadrant 3: Tan was positive
  • Quadrant 4: Cos was positive

Notice the bold first letters? They form the acronym CAST, ... but you must start in the 4th quadrant.
Here's the diagram:

This is called the CAST diagram. All you have to remember is that All the trig functions are positive in quadrant 1, and the diagram tells you what is positive.
For example, the C in quadrant 4 tells you that Cos is positive there; the other two trig fractions are negative.

Let's see if you've got it. Try these questions; scroll up to the CAST diagram to help you answer; the solutions appear below.
  1. In which two quadrants is sin positive?
  2. In which quadrant are sin and cos both negative?
  3. In which quadrants is sin negative?

and the answers are ...

  1. 1 and 2
  2. 3
  3. 3 and 4