I did a question the other day which expected me to know what cos(30) is without a calculator. How do I work it out?

There are certain angles that you can use shapes and trigonometry to work out what their cosine, sine and tangent angles are. These angles are 30, 45 and 60 degrees and there are 2 very simple but important triangles to remember to draw to work them out quickly. 

For cos(30) you need to start with an equilateral triangle (so all the angles of the corners are 60 degrees) with each side length 2, then cut it in half with a line going through one vertex and the middle of the opposite line to make two new triangles. See on the whiteboard how we now have a triangle with side lengths 2, 1 and square root of 3 (using pythagoras), and interior angles 90, 60 and 30 degrees. We can now use the sohcahtoa rule from trigonometry to find that cos(30) is square root of 3 divided by 2. 

I’ll draw this out so you can see how it works on the picture, and let me know if I’ve used any maths that you want me to explain in more detail. You can also use this triangle to get the ones for 60 degrees and I’ll show you the other triangle for the 45 if you want too.