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:
is always positive.
These can be worked out, according to the rule:
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:
Now let's look at an angle in the second quadrant:
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:
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:
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:
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.
and the answers are ...