Question

How do you solve the triangle given m∠C = 13°, m∠A = 22°, c = 9?

Answer 1

Start by drawing a diagram.

We know an angle opposite a side, so we will be using The Law of Sines to solve this problem. Note that this is not an ambiguous case, since we know a side and two angles and not two sides and one angle.

`sinA/a = sinB/b = sinC/c`

`sinA/a = sinC/c`

`(sin22˚)/a = (sin13˚)/9`

Solving for `a`, you should get:

`a ~= 15.0`

We can now use the fact that the angles in every triangle are supplementary, which means that they have a sum of `180˚`, to determine the measure of angle B.

`13˚ + 22˚ + B = 180˚`

`B = 180˚ - 22˚ - 13˚`

`B = 145˚`

We can now use this information to reapply The Law of Sines to determine the length of side `b`.

`sinC/c = sinB/b`

`(sin13˚)/9 = (sin145˚)/b`

Solving for `b`, we obtain `b ~= 22.9`

In summary:

We have solved the triangle. The measures we have found are as follows.

`a~= 15.0" units"`
`B = 145˚`
`b~= 22.9" units"`

Hopefully this helps!