Question

How do you find the remaining side of a `30^circ-60^circ-90^circ` triangle if the longest side is 5?

Answer 1

opposite side`=4.33` , adjacent side `=2.5`

Explanation:

Let the top angle be `color(white)(aaaaaaaa)` `angleA=30°`
Let the bottom left angle be `color(white)(aa)` `angleB=60°`
Let the bottom right angle be `color(white)(a)` `angleC=90°`
AB= hypotenuse side=5 given
BC= adjacent side
AC= opposite side

`:.(BC)/(AB)=Cos 60°`

multiply L.H.S. and R.H.S. by `(AB)/1`

`cancel(AB)/1 xx (BC)/cancel(AB)=(AB)/1 xx Cos60°`

`BC=5 xx cos60°`

`BC=5 xx 0.5`

`BC=2.5`

`(AC)/(AB)=sin60°`

multiply L.H.S. and R.H.S. by `(AB)/1`

`cancel(AB)/1 xx (AC)/cancel(AB)=sin60 xx (AB)/1`

`AC=sin60° xx 5/1`

`AC=0.866025403 xx 5`

`AC=4.330127019`

CHECK:

`tan60°=(AC)/(BC)=(4.330127019)/2.5=arctan1.732050808=60°`

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