Question

A ship travels \(50 \mathrm{~km}\) from \(\mathrm{O}\) on a bearing of \(290^{\circ}\) to get to position A. From A it heads directly to B. Position B is \(90 \mathrm{~km}\) from \(\mathrm{O}\) on a bearing of \(190^{\circ}\). (a) Calculate the distance \(\mathrm{AB}\). (b) Calculate the bearing the ship must follow from \(\mathrm{A}\) to arrive directly at \(\mathrm{B}\).

Short answer

Answer: The distance between points A and B is approximately 86.01 km, and the ship must follow a bearing of approximately 325.51° from point A to arrive directly at point B.

Step-by-step solution

Draw a sketch

Begin by drawing a sketch of the problem to help with visualization. Label points O, A, and B. Draw the bearings and the distances given in the exercise.

Find angle AOB

Since we have the bearings from point O to points A and B, we can calculate the angle AOB. To find the angle AOB, subtract the smaller bearing from the larger one: \(\angle AOB = 290^\circ - 190^\circ = 100^\circ\)

Use the cosine rule to find the distance AB

With the angle AOB and the distances OA and OB, we can apply the cosine rule to find the distance AB: \(AB^{2} = OA^{2} + OB^{2} - 2(OA)(OB)\cos{(\angle AOB)}\) Plugging in the given values: \(AB^{2} = 50^2 + 90^2 - 2(50)(90) \cos{100^\circ}\) Calculate the value of AB: \(AB = \sqrt{2500 + 8100 - 9000 \cos{100^\circ}} \approx 86.01\,\mathrm{km}\) (a) The distance between points A and B is approximately \(86.01\,\mathrm{km}\).

Use the sine rule to find angle OAB

Next, we need to find the angle OAB in order to determine the bearing from A to B. We can use the sine rule since we have the distance AB and two other angles and sides of the triangle: \(\frac{\sin(\angle OAB)}{OA} = \frac{\sin(\angle AOB)}{AB}\) Replacing the values and solving for the angle OAB: \(\angle OAB = \arcsin{\frac{OA \sin{(\angle AOB)}}{AB}}\) \(\angle OAB = \arcsin{\frac{50 \sin{100^\circ}}{86.01}} \approx 35.51^\circ\)

Calculate the bearing from A to B

To find the bearing from A to B, add the angle OAB to the bearing from O to A: Bearing from A to B = Bearing from O to A + Angle OAB Bearing from A to B \(= 290^\circ + 35.51^\circ \approx 325.51^\circ \) (b) The ship must follow a bearing of approximately \(325.51^\circ\) from point A to arrive directly at point B.

Key concepts

Cosine Rule

The Cosine Rule, also known as the Law of Cosines, is a key concept in trigonometry and plays a crucial role in various applications, including navigation. It relates all three sides of a triangle to the cosine of one of its angles. The Cosine Rule is stated as:
\[c^2 = a^2 + b^2 - 2ab\cos(C)\]
where \(a\), \(b\), and \(c\) are the lengths of the sides of a triangle, and \(C\) is the angle opposite side \(c\). This rule is particularly useful when you have either three sides and want to find an angle or two sides and the included angle, allowing you to find the third side.
In the context of navigation, as seen in the given problem, the Cosine Rule helps in determining the distance between two points (A and B) by using the known distances from a common point (O) and the included angle. Here, the rule assists the navigator in calculating the precise distance the ship must travel to reach from point A to point B. It is used when other navigation strategies like Dead Reckoning or Electronic Navigation Systems are not applicable or to confirm findings from those systems.

Sine Rule

The Sine Rule, also known as the Law of Sines, is another fundamental principle in trigonometry, essential for solving various problems, including those in navigation. The rule states that the ratio of the length of a side to the sine of its opposite angle is the same for all three sides of the triangle. The Sine Rule can be expressed as:
\[\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\]
Here, \(a\), \(b\), and \(c\) represent the triangle's sides, while \(A\), \(B\), and \(C\) are the respective opposite angles. The Sine Rule is particularly useful for finding unknown angles or sides when given either two angles and one side or two sides and a non-included angle.
In navigational problems like the one presented, once the distance between two points has been established using the Cosine Rule, the Sine Rule can then be employed to find unknown angles. These angles are often essential for determining the correct bearing, which ensures that the ship follows the intended path.

Bearings Calculation

Bearings are a fundamental concept in navigation, allowing sailors and pilots to identify directions relative to the cardinal points: North, East, South, and West. A bearing is usually expressed in degrees, indicating the direction from one point to another.
To calculate bearings between points, one must understand how to read and interpret them. In navigation, bearings are given as three-digit angles measured clockwise from North (0 or 360 degrees). For instance, a bearing of 290 degrees indicates a direction that is 290 degrees clockwise from North.

Improving Accuracy in Bearings Calculation

For accuracy in calculations, it is crucial to:

• Differentiate between true bearings (referenced to true North) and magnetic bearings (referenced to magnetic North).
• Use precise instruments and ensure they are well-calibrated.
• Keep abreast of magnetic declination changes over time and in different geographical locations.

In the context of the given problem, once the angle OAB is calculated using the Sine Rule, it is added to the original bearing from O to A to ascertain the bearing that the ship must follow to reach point B directly. Knowing how to calculate the bearings and understanding their interpretation is key to safe and effective navigation.