Question

How do you find the remaining side of a `30^circ-60^circ-90^circ` triangle if the side opposite `60^circ` is 6?

Answer 1

Use Trigonometric identities.

Explanation:

Let us assume the side next to `60°` is `a`, and the hypotenuse is `b`.

Then use the Pythagorean theorem.

`b = sqrt(6^2+a^2)`

We know: `" "sin 60° = sqrt3/2`
Then:
`6/sqrt(6^2+a^2)= sqrt3 /2`
`6^2+a^2 = ((6xx2)/sqrt3)^2=48`
`a^2 = 12`
`a = 2sqrt(3)`
`b = 4sqrt(3)`

Answer 2

The side lengths are: `2sqrt3," "6," "4sqrt3`.

Explanation:

The sides of a `30°"-"60°"-"90°` triangle are always of the ratio `1"-"sqrt3"-"2`. Meaning: the side opposite 60° is `sqrt3` times the length of the side opposite 30°, and the side opposite 90° is `2` times as long as the side opposite 30°.

In math:

`"side opposite 60°"/"side opposite 30°"=sqrt3/1=sqrt3`

`"side opposite 90°"/"side opposite 30°"=2/1=2`

We are given the side opposite 60° to be length 6. So, given that the ratio of "the 60° side"-to-"the 30° side" is `sqrt3"-to-1"`, we can solve:

`"side opp. 60°"/"side opp. 30°"=sqrt3`

`6/"side opp. 30°"=sqrt3`

`"        "6/sqrt3"         "="side opp. 30°"`

`"       "(6sqrt3)/3"        "="side opp. 30°"`

`"        "2sqrt3"         "="side opp. 30°"`

And, since "the 90° side" is 2 times as long as "the 30° side", we have

`"side opp. 90°" = 2xx "side opp. 30°"`
`"side opp. 90°" = 2xx 2sqrt 3`
`"side opp. 90°" = 4sqrt 3`.