For 30-60-90 triangle is it x/√3 or x times √3?

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durbanville eNotes educator| Certified Educator

You are referring to some of the "special" angles in trigonometry with the 30-60-90 triangle and then the 45-45-90 triangle.

The sides of these triangles are always in the same ratio and so regardless of the size of the triangle or specifics of the question, the ratios remain the same. The side lengths are `1;2;sqrt3`  . It depends on the question where the `sqrt3` is to be found.

For example, say you need to find `sin 60` . We know that sin is `(opp)/(hyp)` and the hypoteneuse is always the line opposite the right angle. So `sin 60 = sqrt3/2`  as in terms of this triangle 2 is the hypoteneuse.

If you have a question where the `sqrt3` is the denominator you could have `(hyp)/(opp)`  `1/(sin 60)=cosec 60 = 2/sqrt3`   

From special angles we also know that

`cos 30 = (adj)/(hyp) = sqrt3/2`   (Note that it is the same as sin 60! ) So `1/cos 30 = (hyp)/(adj)=sec 30 = 2/sqrt3`   ``

We also know that `tan 30 = (opp)/(adj) =1/sqrt3` and `tan 60 = (adj)/(hyp)=sqrt3/1`

`1/tan60 = (hyp)/(adj)=cot60 = 1/sqrt3`