Question

How to find c? Please read the image and help me please!

Answer 1

`c=54.67`

Explanation:

First, we need to find the value of the missing angle, `γ`. The sum of all angles in a triangle is always `180°`, and we've been given the other two angles, so:

`180°-⍺-β=γ`
`180°-48°-59°=73°`

Let's add that, along with the value of `b`, to the image to make it a bit more clear:

Then, we can either:

1. Solve for the value of `c` directly using the law of sines;
2. or solve for `a`, and then use the law of cosines to solve for `c`.

Obviously, the first way is simpler.
Here's how to do it the first way:

To find the value of `c`, substitute values in for `γ`, `β`, and `b`:

`sinγ/c = sinβ/b`

`(sin73°)/c = (sin59°)/49`

`c=54.67`

Or, we could also do it the second way to use both laws:

To find the value of `a`, substitute values in for `⍺`, `β`, and `b`:

`sin⍺/a = sinβ/b`

`(sin48°)/a = (sin59°)/49`

`a=42.4819`

We now have two sides.

Apply the law of cosines:

`c^2 = a^2 + b^2 - 2*a*b*cosγ`

`c^2 = 42.4819^2 + 49^2 - 2*42.4819*49*cos73°`

`c^2 = 2988.50229`

`c = 54.67`