Problem 52 - A Graphical Approach to Precalculus with Limits

Question

Point \(X\) is \(12.00 \mathrm{km}\) directly west of point \(Y\). From point \(X\), point \(Z\) is on a bearing of \(66^{\circ} 45^{\prime},\) and from point \(Y\) the bearing of \(Z\) is \(336^{\circ} 45^{\prime} .\) (a) Find the distance between \(X\) and \(Z\). (b) Find the distance between \(Z\) and \(Y\).

Step-by-Step Solution

Answer

Distance XZ can be found using the Law of Cosines, and distance ZY can be calculated using the Sine Rule.

Step 1: Understand the Setup

We have three points: X, Y, and Z. X is 12 km directly west of Y. From X, point Z is at a bearing of 66° 45', and from Y, the bearing of Z is 336° 45'.

Step 2: Analyze the Bearings

Convert the bearings to angles relative to the north-south line. Bearing from point X to Z at 66° 45' is already in the correct format. The bearing from Y to Z at 336° 45' translates to an angle of (360° - 336° 45') which is 23° 15' relative to due north.

Step 3: Apply the Law of Cosines (a)

To find the distance XZ, use the Law of Cosines in triangle XYZ. Since XY = 12 km, and the internal angle at Y, formed by lines YX and YZ, is (180° - 66° 45' + 23° 15') = 136° 30', apply the formula: \[XZ^2 = XY^2 + YZ^2 - 2 \cdot XY \cdot YZ \cdot \cos(136° 30')\]Fill in the distance XY = 12 km: \[XZ^2 = 12^2 + YZ^2 - 2 \cdot 12 \cdot YZ \cdot \cos(136° 30')\]

Step 4: Apply Sine Rule to Find ZY

To find ZY, we can use the Sine Rule. In triangle XYZ, angle at X is 23° 15' (bearing adjustment), and we have angle Y = 136° 30'. Use the Sine Rule:\[\frac{ZY}{\sin(66° 45')} = \frac{12}{\sin(136° 30')}\]Solve for ZY:\[ZY = \frac{12 \cdot \sin(66° 45')}{\sin(136° 30')}\]

Step 5: Calculate Distance XZ

From the equation derived in Step 3, solve for XZ once YZ is estimated using a solved or known value. Alternatively, you can simplify calculations for ZY using Step 4's Sin Rule approximation, then substitute to find XZ.

Step 6: Final Calculations

Perform the calculations for each formula used in the previous steps. Ensure all trigonometric calculations are correctly adjusted for the bearings and angles involved.

Key Concepts

Bearing

In navigation and surveying, bearing is the direction or path along which something moves or along which it lies. It is usually measured in degrees from the north direction (0°) in a clockwise manner, up to 360°. For example, a bearing of 0° or 360° points north, 90° points east, 180° points south, and 270° points west. In our exercise, from point X, point Z has a bearing of 66° 45'. This means if you start facing north from X, you would turn 66° 45' clockwise to face point Z.
From point Y, point Z has a bearing of 336° 45', which also means starting north and turning 336° 45' clockwise, which effectively creates an angle of 23° 15' with north if you consider the closest angle back to north.

• Bearings help determine relative positions of points on or above the earth.
• They are crucial for navigation and finding directions in a coordinated system.

Law of Cosines

The Law of Cosines is essential when you need to solve non-right angled triangles. It relates the sides of a triangle to the cosine of one of its angles. In triangle ABC, it is stated as: \[c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\]where \(C\) is the angle opposite side \(c\), and \(a\) and \(b\) are the other two sides.

In our problem, to find the distance between X and Z, we applied the Law of Cosines because we knew:

• Distance XY = 12 km
• The angle at Y, which is 136° 30'

From here, the distance XZ is given by:\[XZ^2 = 12^2 + YZ^2 - 2 \cdot 12 \cdot YZ \cdot \cos(136° 30')\]This formula establishes the relationship between the known and unknown sides of triangle XY and Z.

Sine Rule

The Sine Rule is useful in solving triangles, especially to find unknown distances or angles when certain attributes of the triangle are known. It states: \[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \]where \(a, b, c\) are the sides of the triangle, and \(A, B, C\) are their respective opposite angles.

In this exercise, to find the distance ZY, we use the Sine Rule because we have:

• Angle at Y, 136° 30'
• Angle at X, roughly calculated from bearings, and distance XY = 12 km

The formula becomes:\[\frac{ZY}{\sin(66° 45')} = \frac{12}{\sin(136° 30')}\]This helps us find \(ZY\) using the known elements of the triangle, which are the angles and one side.

Triangles

A triangle is a three-sided polygon and is one of the most fundamental shapes in geometry. It consists of three sides, three angles, and it forms a closed figure. Triangles can be classified based on their sides or angles:

• Equilateral: All sides and angles are equal.
• Isosceles: Two sides are equal, and two angles are equal.
• Scalene: All sides and angles are different.
• Acute: All angles are less than 90°.
• Right: One angle is exactly 90°.
• Obtuse: One angle is greater than 90°.

In the context of trigonometry, triangles are used to solve problems involving distances and angles like in our exercise. In the case of triangle XYZ from the exercise, it is important to note how bearings and geometry help in forming the vertices and sides. Understanding these relationships is crucial in using trigonometric rules effectively to find unknown elements.