Problem 51 - Precalculus with Limits

Question

Flight Path A plane flies 500 kilometers with a bearing of \(316^{\circ}\) from Naples to Elgin (see figure). The plane then flies 720 kilometers from Elgin to Canton (Canton is due west of Naples). Find the bearing of the flight from Elgin to Canton

Step-by-Step Solution

Answer

The bearing of the flight from Elgin to Canton is calculated with the formula \(90^{\circ} - tan^{-1}\left(\frac{720}{500}\right)\) which will give the exact angle in degrees.

Step 1: Understand the Problem

Draw the triangle with Naples (N), Elgin (E) and Canton (C). The bearing of E from N is also \(316^{\circ}\). Due to this, we can conclude that the angle between NE and NC is \(316^{\circ} - 270^{\circ} = 46^{\circ}\), as NC is due west of N. The distances NE and EC have been given as 500 km and 720 km respectively. We need to find the bearing of EC from E.

Step 2: Apply Trigonometric Ratios

We have a right angled triangle, where we know the lengths of two sides. The bearing of EC from E is equal to the value of the angle at E, which can be determined by applying the tangent function. The tangent of an angle in a right-angled triangle is equal to opposite side over adjacent side, in this case that's \(tan(E) = \frac{EC}{NE}\).

Step 3: Calculate Angle E

Substitute the actual lengths (\(tan(E) = \frac{720}{500}\)) and use the inverse tangent function to calculate the angle E. That gives us \(E = tan^{-1}\left(\frac{720}{500}\right)\).

Step 4: Calculate Bearing of EC from E

The bearing of EC from E would be \(90^{\circ} - E\) (due to E being in the North-West quadrant), so we subtract the value of angle E from \(90^{\circ}\) to get the bearing.

Key Concepts

Trigonometric Ratios

Trigonometric ratios are foundational in understanding how to relate the angles and sides of a triangle, especially in right-angled triangles. These ratios include sine, cosine, and tangent, and are dependent on the angles of the triangle.

• **Sine (sin)** of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.
• **Cosine (cos)** is the ratio of the length of the adjacent side to the length of the hypotenuse.
• **Tangent (tan)** is the ratio of the length of the opposite side to the length of the adjacent side.

In navigation and many practical applications, these ratios help determine distances and angles that are not directly measurable. In the given problem, we use the tangent ratio because we know the lengths of two sides of the triangle — namely, the opposite side (720 km) and the adjacent side (500 km). Calculating these ratios gives us an understanding of the angle relationships within the triangle.

Inverse Tangent Function

The inverse tangent function, also known as arctan or \(tan^{-1}\), is used to find the angle whose tangent is a given number. This is particularly helpful in navigation when you know the ratio of two sides of a right-angled triangle but need to find the angle.

In the problem, to find the angle at point E, we used the formula \(E = tan^{-1}\left(\frac{720}{500}\right)\). This involves calculating the ratio 720/500 and identifying the angle whose tangent is that value.

The inverse tangent function is available in most scientific calculators and can sometimes be solved using logarithmic or geometric methods. The angle obtained is then used to determine the bearing or directional heading in navigation.

Right-Angled Triangle

A right-angled triangle is characterized by one angle being exactly 90 degrees. This feature simplifies the use of trigonometric ratios, as the sides opposite and adjacent to the non-right angles can be directly related using sine, cosine, and tangent.

In the context of navigation, a right-angled triangle can represent the path of travel, with different segments of the trip forming the sides of the triangle. In this exercise, the right-angled triangle formed by Naples, Elgin, and Canton allows for a clear application of the tangent ratio.

• One leg represents the north-south distance.
• The other leg represents the east-west distance.
• The hypotenuse is the direct flight path.

Knowing two sides of the triangle (the north-south and east-west distances), we can determine the angle at E using the tangent function.

Navigation

Navigation often involves determining paths and angles of travel between destinations, frequently relying on concepts such as bearings and trigonometry. A bearing represents the direction or path along a compass relative to a fixed point, usually expressed in degrees.

In our problem, the plane's initial direction from Naples to Elgin is given as a bearing of \(316^\circ\), using compass directions where due north is \(0^\circ\) or \(360^\circ\), and due west is \(270^\circ\). This directional measure helps in understanding how a flight path should be adjusted.

When the plane heads from Elgin to Canton, the task is to find the new bearing, using both the calculated angle at E and the understanding that the path is due west relative to Naples. Adjusting by subtracting the angle at E from \(90^\circ\) gives the correct bearing from Elgin to Canton, ensuring accurate and efficient navigation.