Exercises: Sine and Cosine theorem

Given: $a = 5.1 cm$ $b = 8 cm$ $c = 4.3 cm$

Required: $β = ?$

You are given three sides and need to find the value of an angle. Since the angle you are looking for is the angle $β$ , the side standing alone in the cosine theorem is $b$ .

$b^{2}$	$=$	$a^{2} + c^{2} - 2 a c \cdot \cos (β)$	$- a^{2} - c^{2}$
	↓	Solve for $β$ .
$b^{2} - a^{2} - c^{2}$	$=$	$- 2 a c \cdot \cos (β)$	$: (- 2 a c)$
$\frac{b^{2} - a^{2} - c^{2}}{- 2 a c}$	$=$	$\cos (β)$	$\cos^{- 1}$
$β$	$=$	$\cos^{- 1} \frac{b^{2} - a^{2} - c^{2}}{- 2 a c}$
	↓	Plug in the given values.
$β$	$=$	$\cos^{- 1} \frac{8^{2} - {5.1}^{2} - {4.3}^{2}}{- 2 \cdot 5.1 \cdot 4.3}$
$β$	$\approx$	$116.4 °$