Question

How do you use the law of sines to solve the triangle given C = 67°, a = 21, c = 20?

Answer 1

Given two sides of a triangle and a non-contained angle on the longer side, two different triangles are possible.

Casual application of the Sine Law will only give you one of these.
unless you are very careful.

By the Sine Law
`(sin(A))/(21) = (sin(67^o))/20`

`sin(A) = 0.96653`

According to Excel (and I assume any standard calculator)
`arcsin(0.96653) = 75.13436^o`

That is `sin(75.13436^o) = 0.96653`

BUT
`sin(180^o - 75.13436)`
`=sin(104.86564^o)`
`=0.96653` also

So A could be (approx.) `75^o`
or A could be (approx.) `105^o`

and you will need to calculate the third angle and side separately for each of these possibilities