Question

How do you solve the right triangle ABC given A = 61°, a = 17, b = 19?

Answer 1

If ABC is a right triangle
`color(white)("XXXX")`with `A = 61^@`
`color(white)("XXXX")`and `a = 17`
`color(white)("XXXX")`and `b = 19`
It might (at first appear that 2 configurations are possible:

`/_ B = 90^@` (since it is a right triangle)
`/_C = 29^@` (since `/_A+/_B+/_C = 180^@`
and
`c = sqrt(19^2 - 17^2)` (approximately `8.485`)

Unfortunately
Attempting to use the Law of Sines to verify these results:
`color(white)("XXXX")` `(sin A)/a = (sin B)/b = (sin C)/c`

we get
`color(white)("XXXX")` `(sin A)/a = sin(61^@)/17 = 0.051448`
and
`color(white)("XXXX")` `(sin B)/b = sin(90^@)/19 = 0.052362`
and
`color(white)("XXXX")` `(sin C)/c = sin(29^@)/8.485= 0.057135`

Conclusion
The given values are not those of a right triangle!

Answer 2

No solution is possible with the given data.