Question

A tower has a bearing of \(37^{\circ} 00^{\prime}\) when measured from a point \(\mathrm{O}\), and is \(973 \mathrm{~m}\) distant from \(\mathrm{O}\). A chimney has a bearing of \(100^{\circ} 30^{\prime}\) when measured from \(\mathrm{O}\) and is \(1042 \mathrm{~m}\) distant from O. Calculate the distance from the tower to the chimney.

Short answer

Answer: The approximate distance between the tower and the chimney is 577.91 meters.

Step-by-step solution

Convert angles to decimal

We need to convert the given angles from degrees and minutes to decimal degrees: \(37^{\circ}00^{\prime}\) = \(37 + \dfrac{0}{60}\) = \(37^{\circ}\) \(100^{\circ}30^{\prime}\) = \(100 + \dfrac{30}{60}\) = \(100.5^{\circ}\)

Find angle between tower and chimney

We need to find the angle between the tower and the chimney as seen from point O: \(\angle COA= |100.5^{\circ} - 37^{\circ}|= 63.5^{\circ}\)

Use the law of cosines to find distance between tower and chimney

Let \(CA\) be the distance from the tower to the chimney we want to find. We can apply the law of cosines: \(CA^2 = CO^2 + OA^2 - 2 (CO)(OA) \cos{\angle COA}\) Substitute the given values into the equation: \(CA^2 = 1042^2 + 973^2 - 2 \times 1042 \times 973 \times \cos{63.5^{\circ}}\)

Calculate and output the result

Now we can compute the distance from the tower to the chimney: \(CA=\sqrt{1042^2 + 973^2 - 2 \times 1042 \times 973 \times \cos{63.5^{\circ}}} \approx 577.91 \mathrm{~m}\) The distance between the tower and the chimney is approximately \(577.91\) meters.

Key concepts

Law of Cosines

The Law of Cosines is a critical concept in trigonometry, particularly when dealing with non-right triangles. It generalizes the Pythagorean theorem and allows us to calculate the length of any side of a triangle if we know the lengths of the other two sides and the measure of the included angle.

Given a triangle with sides a, b, and c, where c is the side we want to find, and \( \gamma \) is the angle opposite to c, the Law of Cosines formula is:

$$ c^2 = a^2 + b^2 - 2ab \cos(\gamma) $$
In the example provided, we applied the Law of Cosines to find the distance between the tower and the chimney. This calculation assumes that we have a triangle with two known sides (the distances from O to the tower and chimney) and the included angle (difference between bearings).

A common area of confusion can arise if the student does not properly identify the included angle. Remember, the included angle is the angle between the two known sides—we calculate this by finding the difference in bearings. Getting this step right is paramount since inaccurate angle measurements lead to incorrect solutions.

While sometimes daunting, the Law of Cosines unlocks the power to solve a wide array of problems and is a fundamental tool in fields like navigation, engineering, and physics.

Bearing Calculations

Bearing calculations are used to describe the direction or position of one point in relation to another, typically in navigation and surveying. Bearings are usually measured in degrees from North, with a clockwise direction. It is the angular measurement between two points with the observer's position as the vertex.

Understanding how to work with bearings is essential when solving trigonometry problems related to navigation. When the problem involves finding the distance between two points with given bearings from a third point, as in our exercise, we calculate the angle between those points. The difference in bearing, subtracted from one another, gives us this angle. It is essential, though, to ensure that the bearings are measured from the same reference point and in the same direction to get an accurate angle for further calculations.

In practice, real-life navigators and surveyors rely heavily on such calculations to traverse vast distances and accurately pinpoint locations. While bearing calculations might seem complex, mastering them can be incredibly rewarding and practical.

Angle Conversion

In trigonometry, angle measurements can be represented in either degrees, minutes, and seconds (DMS) or decimal degrees. Engineers, mathematicians, and scientists often need to convert between these two systems for calculations and analysis. The step-by-step example showcases this through the conversion of the angle from degrees and minutes into decimal degrees before it can be used in the Law of Cosines formula.

In converting DMS to decimal degrees, follow this simple method:

• Determine the total number of minutes and divide by 60 to convert to degrees.
• Add this value to the degree measure to get the angle in decimal degrees.

The conversion step is crucial to ensuring mathematical accuracy in the Law of Cosines and many other trigonometric formulas. A conversion error could lead to incorrect results and possibly a flawed understanding of subsequent concepts, showing the importance of attention to detail in every aspect of trigonometry. Understanding angle conversions empowers students to tackle problems across various areas where precise angle measurements are necessary, such as astronomy, cartography, and computer graphics.