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Exact trig ratios

I am studying Further maths and reading Further Pure Maths, OUP and there is an example in complex numbers section (complex numbers not relevant to my question) where it is basically a trig/vector sort of question.

There is a right angle triangle with opposite side 2 root(3) and adjacent side 2. OK, I can divide by 2 to get opp = root(3) and adj = 1.

Getting hypotenuse is easy, 4.

Working out the angle between x axis and the hypotenuse is easy enough - tan theta = 60 degrees. But the thing that got me in the example was that it just has:

tan^{-1} \theta = \frac{\pi}{3}

I do get some of the standard ratios, eg I can draw a triangle with eg sides 1 x 1 and hyp = root(2). Another one I know is hyp = 2 and adj = 1, therefore, opp = root(3). But tan root(3) seems a bit harder to me. Am I supposed to know this angle?

I can put numbers into calculator and see what I get. But with my calculator in radians mode you sometimes get a decimal number and it is hard to translate to the exact root or fraction. Am I supposed to memorise these?

Just looking for tips on how to work out exact trig ratios. Do you just use the calculator. How do you get to an exact answer in an elegant way?

Reply 1

ztibor

Original post by acomber

tan^{-1} \theta = \frac{\pi}{3}

In the first quadrant you can work out the exact value of degree 0, 30, 45, 60 and 90
( an then 120, 135, 150, 180 .... and so on)

For more values you can use trig equations:
1. Half angle:
e.g sin^2 ß/2=(1-cosß)/2 -> degree 15
1. Addition rules

e. g sin(ß1+ß2)=sinß1*cosß2+cosß1*sinß2 ß1=60, ß2=15 -> degree 75