Question

How do you find the missing sides of a triangle given two angles and a side given Sides are x, y, and 10, two of the angles are 40 and 90?

Answer 1

If `10` = hypotenuse: `x~~6.43, y~~ 7.66`
If `10` is across from `50^@, x~~ 8.39, y ~~ 13.05`
If `10` is across from the `40^@, x~~ 11.92, y~~ 15.56` bolded text

Explanation:

Given: A triangle has two angles: `40^@, 90^@`, and side lengths `x, y, & 10`.

Since we have a `90^@` angle you know that you have a right triangle, which means you can use trigonometric functions.

There are three possible cases:

The longest side is always across from the largest angle. This means that `10` is the hypotenuse if #10 is the longest side.

1. Assume `10` is the hypotenuse:

Since a triangle's angles sum to `180^@`, the third angle is `50^@`.

`cos 40^@ = y/10; " " y = 10 cos 40^@ ~~ 7.66`

`sin 40^@ = x/10; " " x = 10 sin 40^@ ~~ 6.43`

1. Assume `10` is across from the `50^@` angle:

`tan 40^@ = x/10; " " x = 10 tan 40^@ ~~ 8.39`

`cos 40^@ = 10/y; " " y = 10/(cos 40^@) ~~ 13.05`

1. Assume `10` is across from the `40^@` angle:

`tan 40^@ = 10/x; " "x = 10/(tan 40^@) ~~ 11.92`

`sin 40^@ = 10/y; " " y = 10/(sin 40^@) ~~ 15.56`