# How do you solve the triangle given a=1.42, b=0.75, c=1.25?

May 16, 2017

A = 86.68
B = 31.82
C = 61.5
(2dp)

#### Explanation:

Just to clarify, $A$ is the angle opposite side $a$, $B$ opposite side $b$, and the same for $C$.

1st step : Get your calculator!

2nd step : Let's start by working out the angle $A$. The Law of Cosines states that $\cos A = \frac{{b}^{2} + {c}^{2} - {a}^{2}}{2 b c}$
Let's substitute the known values :
$\cos A = \frac{{0.75}^{2} + {1.25}^{2} - {1.42}^{2}}{2 \cdot 0.75 \cdot 1.25}$

3rd step : At this point, you will probably need a calculator. Ok, let's simplify this. With my calculator, I got : $\frac{181}{3125}$

4th step : So right now, $\cos A = \frac{181}{3125}$
We need to get $A$ on its own. To do that, we take the inverse cosine of $\frac{181}{3125}$.
${\cos}^{-} 1 \left(\frac{181}{3125}\right) = 86.67957016 \ldots \ldots$
Let's just round that to 2 decimal places : 86.68

Next, let's work out side B. Although you can solve this using sine, since you asked us about using the Law of Cosine, I will use the Law of Cosine.

$\cos B = \frac{{a}^{2} + {c}^{2} - {b}^{2}}{2 a c}$

Substitute known values :

$\cos B = \frac{{1.42}^{2} + {1.25}^{2} - {0.75}^{2}}{2 \cdot 1.42 \cdot 1.25}$

Plug in numbers on your calculator

$\cos B = \frac{7541}{8875}$

Take the inverse cosine of that :

$B = 31.82201662$
Round it to 2 decimal places : 31.82

Ok, so because interior angles add up to 180 in a triangle, let's set this equation for the last angle :

$31.82 + 86.68 + C = 180$
$118.5 + C = 180$
$C = 61.5$

I'll leave it for you to check angle C using the Law of Cosines.