Sine Rule and Cosine Rule

The three sides of a triangle are given: $a =10 \mathrm{cm}, b =8 \mathrm{cm}, c =7 \mathrm{cm}$. Find the angle $C$.

In a triangle $ABC$, if $a=2, b=3$ and $\sin A=\frac{2}{3}$, find $\angle B$.

An aircraft tracking station spots two aircraft. It determines the distance from a common point $O$ to each aircraft and the angle between the aircraft. If the angle between the two aircraft is $52°$ and the distances from point $O$ to the two aircraft are $58 \mathrm{km}$ and $75 \mathrm{km}$, find the distance between the two aircraft. Write down your answer correct to $1$ decimal place. [Enter the value excluding units]

Pam is researching old steam engines. The following diagram shows a representation of a mechanism she is exploring.

$AB$ is fixed at $2.4 \mathrm{m}$ and $BD$ is fixed at $7.7\mathrm{m}$. $D$ can move only horizontally back and forth as $B$ moves anticlockwise around the circle. $AD$ is the distance from the point $D$ to the centre of the circle, $\mathrm{\alpha }$, is the angle $\mathrm{D}\hat{\mathrm{A}}\mathrm{B}$.

The following data is collected.

Hence or otherwise comment critically on the fit of your model.

Pam is researching old steam engines. The following diagram shows a representation of a mechanism she is exploring.

$AB$ is fixed at $2.4 \mathrm{m}$ and $BD$ is fixed at $7.7\mathrm{m}$. $D$ can move only horizontally back and forth as $B$ moves anticlockwise around the circle. $AD$ is the distance from the point $D$ to the centre of the circle, $\mathrm{\alpha }$, is the angle $\mathrm{D}\hat{\mathrm{A}}\mathrm{B}$.

The following data is collected.

Calculate $AD$ in terms of the angles $\alpha$ and $\beta$ and the lengths of $AB$ and $BD$, using the cosine rule.

To find the third side of $\Delta ABC$ with $\mathrm{AB}=40\mathrm{cm},\mathrm{BC}=25\mathrm{cm}$ and $B\hat{A}C=35°$ Velina and Kristian offered the following suggestions: Kristian suggested: "Use the cosine rule as you are given two sides and one angle." Velina suggested: "Use the sine rule as you are given two sides and an angle opposite to one of them." State whose method is correct and justify your statement. Hence, find third side, $AC$ of the triangle .

The straight-line distance between two cities $A$ and $B$ is $223 \mathrm{km}$. The straight-line distance between cities $A$ and $C$ is $152 \mathrm{km}$. The straight-line distance between cities $B$ and $C$ is $285 \mathrm{km}$. Find the angles between each of the three cities.

A hiker leaves camp and walks $5 \mathrm{km}$ on a bearing of $058°.$ He stops for a break, then continues walking for another $8 \mathrm{km}$ on a bearing of $103°.$ He stops again before heading straight back. Find the distance back to camp from the second stop.

A surveyor has to determine the distance between points $A$ and $B$ that lie on opposite banks of a river. Another point $C$ is $450 \mathrm{m}$ from $A$, on the same side of the river, such that $\angle B\stackrel{}{A}C=45°$ and $\angle ACB=55°$. Find the distance from $A$ to $B$ to the nearest metre.

Use the given information to find all sides and angles in the following triangle.

Use the given information to find all sides and angles in the following triangle.

Use the given information to find all sides and angles in the following triangle.

Use the given information to find all sides and angles in the following triangle.

$c=22 \mathrm{m}, \hat{A}=25°\text{ and }\hat{B}=83°$

Use the given information to find all sides and angles in the following triangle.

$a=11\mathrm{mm},b=14\mathrm{mm}\text{ and }c=18\mathrm{mm}$

Use the given information to find all sides and angles in the following triangle.

$a=12 \mathrm{cm},b=6.5 \mathrm{cm}\text{ and }c=7 \mathrm{cm}$

A ship is sailing due north. The captain sees a lighthouse $12\mathrm{km}$ away on a bearing of $36°$. Later, the captain observes that the bearing of the lighthouse is $125°.$ How far did the ship travel between these two observations?

In $\Delta ABC,AC=8 \mathrm{cm},\hat{C}=101°\text{ and }\hat{B}=32°$. Find the lengths of $AB \& BC$.

Find the unknown angle $\hat{C}$ in the following $△ABC$, where $a=7.2 \mathrm{m} , c=5.1 \mathrm{m}$ and $\hat{A}=70°$. Give your answer correct to $3$ significant figures.

Find the unknown angle $\hat{B}$ in the following $△ABC$, where $b=42.8 \mathrm{cm}, c=30.6 \mathrm{cm}$ and $\hat{C}=41°$. Give your answer correct to $3$ significant figures.

Find the unknown angle $\hat{A}$ in the following $△ABC$, where $a=12.5 \mathrm{cm} , b=15.4 \mathrm{cm}$ and $\hat{B}=68°$. Give your answer correct to $3$ significant figures.