Math Concepts Explained

The values of trig functions of specific angles can be represented by known ratios, and are good to commit to memory to help you work through problems faster. These 'special' angles can be remembered by examining two different triangles. Specifically, the trig functions for angles of 0, 30, 45, 60, 90 degrees are the special ones.

Let's look at the first triangle, and this will hopefully become clear. Take a right angle triangle with two 45 degree angles, and with sides of 1 unit length. By the Theorem of Pythagoras, the hypotenuse of this triangle is of length √2. This is what this triangle looks like:
So then, from these values and SOHCAHTOA, you can obtain the trig values for this special angle of 45 degrees. You can see that:

Sin(45) = 1/√2
Cos(45) = 1/√2
Tan(45) = 1

Don't worry if you can't remember these exact ratios... the simplest thing to remember is how to construct the triangle... which is as easy as remembering a right angle triangle with a 45 degree angle and 2 sides of length 1... you can easily fill in the rest, and then work out the trig ratios yourself!

The second triangle to remember is slightly more complex, but still straightforward. Take a right angle triangle with angles of 30 and 60 degrees. The simplest lengths of the sides of this triangle are 1, 2, √3 (with 2 being the longest side, the hypotenuse). This triangle looks like this:
So then, using the these values, you can obtain the trig values for these special angles as well:

Sin(30) = 1/2
Cos(30) = √3/2
Tan(30) = 1/√3

Sin(60) = √3/2
Cos(60) = 1/2
Tan(60) = √3/1 = √3

Again, just remember how to construct the triangle, and the ratios are easy to come up with!

For 0 and 90 degrees, there isn't a simple triangle scheme to remember the values (although please feel free to correct me if I am wrong!). However, these aren't scary square root numbers or weird fractions:

Sin(0) = 0
Cos(0) = 1
Tan(0) = 0

Sin(90) = 1
Cos(90) = 0
Tan(90) = undefined

If you can familiarize yourself with all of these special angles, or at least understand how to derive them, then you will have a much easier time working on trigonometry questions.