Math Geometry Sine, Cosine and Tangent Sine and Cosine theorem Exercises: Sine and Cosine theorem

Exercises: Sine and Cosine theorem

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 $a^{2} = \frac{1}{4} (e^{2} + f^{2}) - \frac{e \cdot f}{2} \cdot \cos (ε)$ :

$\begin{array}{lrcll} 36 & \approx & 50 - 47.273 \cos (ε) & | - 50 \\ \Leftrightarrow & - 14 & \approx & - 47.273 \cos (ε) & | : (- 47.273) \\ \Leftrightarrow & \cos (ε) & \approx & 0.2962 \\ \Leftrightarrow & ε & \approx & 72.77 ° \end{array}$

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