Question

How do you solve the triangle given m∠B = 105°, b = 23, a = 14?

Answer 1

Start by drawing a diagram.

Since we have one angle with an opposite side and another side known, we are dealing with an ambiguous case. There are either `0, 1 or 2` solutions to this triangle.

We start by determining the measure of `A` using the law of Sines, since we already know the measure of side `a`.

`sinA/a = sinB/b`

`sinA/14 = (sin105˚)/23`

`A = arcsin((14sin105˚)/23)`

`A = 36˚`

There is only one solution to this triangle, because if it was, then the alternative measure of A would be `180˚ - 36˚ = 144˚`, which when added to `B`, would make the sum of the angles in the triangle exceed `180˚`.

We can now use the measure of angles `A and B` to solve for angle C.

`A + B + C = 180˚`

`36˚ + 105˚ + C = 180˚`

`C = 39˚`

The last step is using this information to reapply the law of sines and determine the length of side `c`.

`sinB/b = sinC/c`

`(sin105˚)/23 = (sin39˚)/c`

`c = (23sin39˚)/(sin105˚)`

`c= 15`

In summary

The triangle has the following measures.

`•A = 36˚`
`•a = 14" units"`
`•B = 105˚`
`•b = 23" units"`
`•C = 39˚`
`•c = 15" units"`

Hopefully this helps!