Is there an easy way to remember the unit circle?

Question

It's so hard to remember the values of sine and cosine from 0^o to 90^o (0 to π/2 radians). Looking for an easy way to remember it!

Answer 1

The standard way that the trig functions are taught in the first quadrant is in a reduced form. This is nice for simplifying answers, but a nightmare for memorization. It's usually presented like this:

θ | sin(θ) | cos(θ)

0 | 0 | 1

30 | 1/2 | √3/2

45 | √2/2 | √2/2

60 | √3/2 | 1/2

90 | 1 | 0

BLECH. There's no pattern! Here's the way I teach my students to remember the chart.

θ | sin(θ) | cos(θ)

0 | √0/2 | √4/2

30 | √1/2 | √3/2

45 | √2/2 | √2/2

60 | √3/2 | √1/2

90 | √4/2 | √0/2

What a beautiful pattern! To get tangent of an angle, just divide sine by cosine. Since they all share the same denominator, those denominators cancel. For example, tan(60) = √3/2 / √1/2. The 2's cancel, leaving √3 / √1 or just √3. The 2's will ALWAYS cancel for tangent of these angles. The only one to be careful with is tan(90), which is undefined since you can't divide by zero!

Why should teachers bother explaining these "special" triangles if they're not even going to show what's so special about them? They're special because of this pattern, not because we can write the sine and cosine explicitly -- other angles let you do that, e.g. sin(18) = (√5 - 1)/2.

Hope this helps!

~Logan

Answer 2

Its as easy as 1,2,3

Start with sin Sin30= 1/2

Sin45 sq root2/2

Sin60 sqroot3/2

Note that the denominator for each of these is 2 and the numerator goes from 1 to 2 to 3

Cosine is the exact opposite starting at 30 denominator again is always 2 but the numerator goes from 3 to 2 to 1