There is a pattern which is quite common:
This pattern is useful because it helps us to memorize the trig functions of the main angles. However, when we start to look for the sine of some angles like or , we may struggle for some time. However, there is a beautiful pattern:
That's amazing, isn't it? But I want to tell you, there's a reason behind. I'll show you that.
Firstly, from the first pattern, we know that:
Then, we'll use the half-angle formula
Therefore, we get the the result below:
I am so proud of myself that I can prove this beautiful pattern! (I first found this pattern when I see this) I hope it will help you guys when you are doing some work about trigonometry.