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Using Bearings to Find Distance

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Using Bearings to Find Distance 1

Using Bearings to Find Distance 1

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Question 1 of 4

Find the distance the speedboat has traveled due south

$(AB)$

Round your answer to two decimal places

(164.54) $\text(km due South)$

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Sin Ratio

$sin=\frac{\color{#004ec4}{\text{opposite}}}{\color{#e85e00}{\text{hypotenuse}}}$

Cos Ratio

$cos=\frac{\color{#00880a}{\text{adjacent}}}{\color{#e85e00}{\text{hypotenuse}}}$

Tan Ratio

$tan=\frac{\color{#004ec4}{\text{opposite}}}{\color{#00880a}{\text{adjacent}}}$

A true bearing is an angle measured clockwise from the True North around to the required direction.

First, find the value of angle $CAB$ .

Notice that angle

$CAB$ is part of the given bearing, which also consists of a straight angle from the North line clockwise to the South line.

To find the value of angle

$CAB$ , simply subtract the measure of the straight angle,

$180°$ , from the given bearing.

$/_CAB$	$=$	$210°-180°$
	$=$	$30°$

To solve for line $AB$ , we can use the known values of the hypotenuse and angle $CAB$ .

Use

$cos$ to find the value of line

$AB$ .

$cos$ $30°$	$=$	$\frac{\color{#00880A}{AB}}{\color{#e85e00}{\text{hypotenuse}}}$

$cos$ $30°$	$=$	$\frac{\color{#00880A}{AB}}{\color{#e85e00}{\text{190}}}$

$cos30°$ $xx190$	$=$	$((AB)/190)$ $xx190$	Multiply both sides by $190$

$190cos30°$	$=$	$AB$

Using your calculator,

$190cos30°=164.54$ .

Therefore, the speedboat is

$164.54 \text(km)$ due South.

$164.54 \text(km due South)$

Question 2 of 4

Find the distance (

$d$ ) the boat has traveled from the port (

$P$ )

Round your answer to the nearest kilometre

$d=$ (25) $\text(km)$

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Sin Ratio

$sin=\frac{\color{#004ec4}{\text{opposite}}}{\color{#e85e00}{\text{hypotenuse}}}$

Cos Ratio

$cos=\frac{\color{#00880a}{\text{adjacent}}}{\color{#e85e00}{\text{hypotenuse}}}$

Tan Ratio

$tan=\frac{\color{#004ec4}{\text{opposite}}}{\color{#00880a}{\text{adjacent}}}$

A true bearing is an angle measured clockwise from the True North around to the required direction.

To solve for line $d$ , we can use the known values of the line adjacent to the bearing.

Use

$cos$ to find the value of line $d$ .

$cos48°$	$=$	$\frac{\color{#00880A}{\text{adjacent}}}{\color{#e85e00}{d}}$

$cos48°$	$=$	$\frac{\color{#00880A}{17}}{\color{#e85e00}{d}}$

$cos48°$ $xxd$	$=$	$((17)/d)$ $xxd$	Multiply both sides by $d$

$dcos48°$	$=$	$17$

$dcos48°$ $divcos48°$	$=$	$17$ $divcos48°$	Divide both sides by $cos48°$

$d$	$=$	$17/(cos48°)$	Divide both sides by $cos48°$

Using your calculator,

$17/(cos48°)=25.46$ .

Rounding it to the nearest kilometre, the boat traveled

$25 \text(km)$ from the port.

$25 \text(km)$

Question 3 of 4

A ship has traveled

$72 \text(km)$ North West from the port (

$O$ ). How far North (

$y_n$ ) of the starting point has it traveled?

Round your answer to one decimal place

$y_n=$ (50.9) $\text(km)$

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Sin Ratio

$sin=\frac{\color{#004ec4}{\text{opposite}}}{\color{#e85e00}{\text{hypotenuse}}}$

Cos Ratio

$cos=\frac{\color{#00880a}{\text{adjacent}}}{\color{#e85e00}{\text{hypotenuse}}}$

Tan Ratio

$tan=\frac{\color{#004ec4}{\text{opposite}}}{\color{#00880a}{\text{adjacent}}}$

A true bearing is an angle measured clockwise from the True North around to the required direction.

To solve for the line $y_n$ , we can use the known values of the hypotenuse and the given angle

$(45°)$ .

Remember

$NW$ (North West) means

$45°$ .

Use

$cos$ to find the value of line $y_n$ .

$cos45°$	$=$	$\frac{\color{#00880A}{{y_{n}}}}{\color{#e85e00}{\text{hypotenuse}}}$

$cos45°$	$=$	$\frac{\color{#00880A}{{y_{n}}}}{\color{#e85e00}{72}}$

$cos45°$ $xx72$	$=$	$((y_n)/72)$ $xx72$	Multiply both sides by $72$

$72cos45°$	$=$	$y_n$
$y_n$	$=$	$72cos45°$

Using your calculator,

$72cos45°=50.9$ .

Therefore, the boat traveled

$50.9 \text(km)$ to the North.

$y_n=50.9 \text(km)$

Question 4 of 4

A ship has traveled

$51 \text(km)$ from the port (

$P$ ) with a bearing of

$N 67°E$ . How far east (

$x_e$ ) has it traveled?

Round your answer to one decimal place

$x_e=$ (46.9) $\text(km)$

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Sin Ratio

$sin=\frac{\color{#004ec4}{\text{opposite}}}{\color{#e85e00}{\text{hypotenuse}}}$

Cos Ratio

$cos=\frac{\color{#00880a}{\text{adjacent}}}{\color{#e85e00}{\text{hypotenuse}}}$

Tan Ratio

$tan=\frac{\color{#004ec4}{\text{opposite}}}{\color{#00880a}{\text{adjacent}}}$

A true bearing is an angle measured clockwise from the True North around to the required direction.

To solve for $x_e$ , we can use the known values of the hypotenuse and the given angle,

$(67°)$ .

Use

$sin$ to find the value of line

$x_e$ .

$sin67°$	$=$	$\frac{\color{#004ec4}{x_e}}{\color{#e85e00}{\text{hypotenuse}}}$

$sin67°$	$=$	$\frac{\color{#004ec4}{x_e}}{\color{#e85e00}{51}}$

$sin67°$ $xx51$	$=$	$((x_e)/51)$ $xx51$	Multiply both sides by $51$

$51sin67°$	$=$	$x_e$
$x_e$	$=$	$51sin67°$

Using your calculator,

$51sin67°=46.9$ .

Therefore, the boat traveled

$46.9 \text(km)$ to the East.

$x_e=46.9 \text(km)$