# How do you find the measure of each of the angles of a triangle given the measurements of the sides are 12, 20, 22?

Dec 6, 2016

Use the Law of Cosines as demonstrated below.

#### Explanation:

Let $\angle A$ be the angle opposite side $\left\mid a \right\mid = 22$
and $\angle B$ be the angle opposite side $\left\mid b \right\mid = 20$
and $\angle C$ be the angle opposite side $\left\mid c \right\mid = 12$

The Law of Cosines tells us that
$\textcolor{w h i t e}{\text{XXX}} {\left\mid a \right\mid}^{2} = {\left\mid b \right\mid}^{2} + {\left\mid c \right\mid}^{2} - 2 \left\mid b \right\mid \left\mid c \right\mid \cos \left(A\right)$
or (in a form more useful for this problem):
$\textcolor{w h i t e}{\text{XXX}} \cos \left(\angle A\right) = \frac{{\left\mid b \right\mid}^{2} + {\left\mid c \right\mid}^{2} - {\left\mid a \right\mid}^{2}}{2 \left\mid b \right\mid \left\mid c \right\mid}$
or...
color(white)("XXX")/_A = "arccos"((abs(b)^2+abs(c)^2-abs(a)^2)/(2abs(b)abs(c)))

Using the given values (and a calculator)
$\textcolor{w h i t e}{\text{XXX}} \angle A \approx 1.445468$ (radians)

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Similarly, we can find:
$\textcolor{w h i t e}{\text{XXX}} \angle B \approx 1.124289$ (radians)
and
$\textcolor{w h i t e}{\text{XXX}} \angle C \approx 0.571835$ (radians)