Exercises: Sine and Cosine theorem

1

For this task you need the following basic knowledge: Cosine theorem
Given: $a = 4 cm$ $b = 6 cm$ $γ = 67^{\circ}$
Required: $c = ?$
Use the cosine theorem.
$c^{2}$ $=$ $a^{2} + b^{2} - 2 a b \cdot \cos (γ)$
↓
Plug in the given values.
$c^{2}$ $=$ $4^{2} + 6^{2} - 2 \cdot 4 \cdot 6 \cdot \cos (67^{\circ})$
↓
Add up the right-hand side.
$c^{2}$ $\approx$ $33.24$ $\sqrt{}$
$c$ $=$ $5.77$

5
The sketch shows a parallelogram with side lengths $a = 6 cm$ and $b = 8 cm$ , as well as the angle $α = 70^{\circ}$ .
Calculate the angle $ε$ .

For this task you need the following basic knowledge: Properties of a parallelogram
In particular, you need the following rules:
The diagonals in a parallelogram bisect each other.
Adjacent angles add up to $180^{\circ}$ .
Set up target equation
First we look for an equation that contains the angle $ε$ we are looking for. In the parallelogram, only the side lengths $a$ and $b$ and the angle $α$ are known. Since the side a is opposite the angle we are looking for, the cosine theorem for $a$ is a good choice.
Since the diagonals bisect each other in a parallelogram, the sides adjacent to the angle $ε$ have the length $\frac{e}{2}$ and $\frac{f}{2}$ respectively. The objective equation is therefore:
Simplify the target equation
To have to calculate less later, you can simplify the equation by factoring it out and shortening it
$a^{2} = \frac{1}{4} (e^{2} + f^{2}) - \frac{e \cdot f}{2} \cdot \cos (ε)$
Calculate missing quantities
You can calculate the lengths of $e$ and $f$ with the cosine theorem:
$e^{2} = a^{2} + b^{2} - 2 a b \cos (α)$
$f^{2} = a^{2} + b^{2} - 2 a b \cos (β)$
In a parallelogram, adjacent angles add up to $180 °$ . So $β = 180 ° - α$ . Together with the supplement relations for sine and cosine, the following applies:
$\begin{array}{lr} e^{2} = 100 - 96 \cdot \cos (α) \approx 67.17; & e \approx 8.20 \\ f^{2} = 100 + 96 \cdot \cos (α) \approx 132.83; & f \approx 11.53 \end{array}$
Insertion into the target equation
Insertion into the target equation $a^{2} = \frac{1}{4} (e^{2} + f^{2}) - \frac{e \cdot f}{2} \cdot \cos (ε)$ :
$\begin{array}{lrcll} 36 & \approx & 50 - 47.273 \cos (ε) & | - 50 \\ \Leftrightarrow & - 14 & \approx & - 47.273 \cos (ε) & | : (- 47.273) \\ \Leftrightarrow & \cos (ε) & \approx & 0.2962 \\ \Leftrightarrow & ε & \approx & 72.77 ° \end{array}$

$\frac{b}{\sin (β)}$	$=$	$\frac{c}{\sin (γ)}$	$\cdot \sin (γ)$
	↓	Hint: By forming the reciprocal value of both fractions, you can get the variable you are looking for into the numerator.
$\frac{\sin (β)}{b}$	$=$	$\frac{\sin (γ)}{c}$	$\cdot b$
	↓	Solve for the required variable.
$\sin (β)$	$=$	$\frac{\sin (γ)}{c} \cdot b$
	↓	Plug in the given values.
$\sin (β)$	$=$	$\frac{\sin (108^{\circ})}{7.5} \cdot 4$
$\sin (β)$	$\approx$	$0.5072$	$\sin^{- 1}$
$β$	$\approx$	$30 .48 °$

$b^{2}$	$=$	$a^{2} + c^{2} - 2 a c \cdot \cos (β)$	$- a^{2} - c^{2}$
	↓	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.
$β$	$=$	$\cos^{- 1} \frac{8^{2} - {5.1}^{2} - {4.3}^{2}}{- 2 \cdot 5.1 \cdot 4.3}$
$β$	$\approx$	$116.4 °$

$c^{2}$	$=$	$a^{2} + b^{2} - 2 a b \cdot \cos (γ)$
	↓	Plug in the given values.
$c^{2}$	$=$	$4^{2} + 6^{2} - 2 \cdot 4 \cdot 6 \cdot \cos (67^{\circ})$
	↓	Add up the right-hand side.
$c^{2}$	$\approx$	$33.24$	$\sqrt{}$
$c$	$=$	$5.77$

$\frac{a}{\sin (α)}$	$=$	$\frac{c}{\sin (γ)}$	$\cdot \sin (γ)$
	↓	Solve for the variable you are looking for. To do this multiply by $\sin (γ)$ .
$c$	$=$	$\frac{a}{\sin (α)} \cdot \sin (γ)$
	↓	Plug in the values.
$c$	$=$	$\frac{9}{\sin (94^{\circ})} \cdot \sin (61^{\circ})$
$c$	$\approx$	$7.89$

Exercises: Sine and Cosine theorem

Set up target equation

Simplify the target equation

Calculate missing quantities

Insertion into the target equation