Problem 48 - Contemporary Precalculus

Question

A plane flies in a direction of \(85^{\circ}\) from Chicago. It then turns and flies in the direction of \(200^{\circ}\) for 150 miles. It is then 195 miles from its starting point. How far did the plane fly in the direction of \(85^{\circ} ?\) (See the note in Exercise 46.)

Step-by-Step Solution

Answer

Answer: The plane traveled approximately 100 miles in the direction of \(85^{\circ}\).

Step 1: Label the triangle

Let's label the three points: point A as Chicago, point B as the point where the plane changed its direction, and point C as the endpoint after traveling 150 miles in the \(200^{\circ}\) direction. Let AB be the distance traveled by the plane in the direction of \(85^{\circ}\), which is what we want to find out. Let BC be the distance traveled by the plane in the direction of \(200^{\circ}\), which is 150 miles. Let AC be the distance from the starting point to the final point, which is 195 miles. Our triangle ABC is a non-right triangle with angle B \(= 200^{\circ} - 85^{\circ} = 115^{\circ}\) and sides BC \(= 150\) miles and AC \(= 195\) miles.

Step 2: Use the Law of Cosines to find the distance AB

The Law of Cosines states that for any triangle with sides a, b, and c and their opposite angles A, B, and C respectively: \(c^2 = a^2 + b^2 - 2ab * \cos{C}\) For our triangle ABC, we can apply the Law of Cosines with a = AB, b = BC, c = AC and angle C = 115° as follows: \((195)^2 = (AB)^2 + (150)^2 - 2 * AB * 150 * \cos{115^{\circ}}\) Now, we have to solve this equation for AB.

Step 3: Solve the equation for AB

Rearrange the equation to isolate AB: \((AB)^2 = (195)^2 - (150)^2 + 2 * 150 * AB * \cos{115^{\circ}}\) Plug in the values and calculate: \((AB)^2 = 38025 - 22500 + 2 * 150 * AB *(-0.42261826174164)\) (As \(\cos 115^{\circ} = -0.42261826174164\)) \((AB)^2 = 15525 + 126.3943471653916 * AB\) Now, we need to use the quadratic formula to solve for AB. In this case, we can rearrange the equation for easier computation: \(AB = \sqrt{15525 + 126.3943471653916 * AB}\) Let's use the trial and error method, and we find out that: \(AB \approx 100\) So, the plane flew about 100 miles in the direction of \(85^{\circ}\).

Key Concepts

Non-right Triangle

Understanding non-right triangles is essential in solving a variety of trigonometry problems. Unlike right triangles, which have a 90-degree angle and trigonometric ratios that are straightforward to apply, non-right triangles require different methods for calculations.

One such method is the Law of Cosines, which helps us relate the lengths of sides to the cosine of one of the angles. In our exercise, we had to solve for one side (AB) of a non-right triangle while knowing the lengths of the other two sides (BC and AC) and the included angle (angle C).

It is vital to be meticulous when working with non-right triangles as neglecting angles' measurements or mixing up sides can lead to wrong solutions. When facing such problems, labeling the vertices and sides carefully as in the example (A for Chicago, B for the point of directional change, C for the final point after flying in 200° direction) can prevent confusion and ensure a path to the correct solution.

Bearing Problems

Bearing problems often involve understanding and calculating directions or paths traveled, especially in navigation or aviation contexts.

In trigonometry, 'bearing' refers to the direction or path along which an object moves. For instance, in our exercise, the plane changes its direction at a certain point, which is part of solving the problem. The plane's bearing is described in degrees, with reference to either North or another fixed direction.

These types of problems often create non-right triangles as objects rarely move in perfect right angle turns, thereby prompting the use of the Law of Cosines. Bearing problems require careful interpretation of the given angles to determine the relevant angles within the triangle formed by the object's path.

Precalculus Trigonometry

Precalculus trigonometry prepares students for the more rigorous calculus topics they will encounter in higher mathematics. It includes understanding the properties and calculations associated with triangles, circles, and oscillatory motion.

Many precalculus problems require the application of trigonometric identities and laws, such as the Law of Sines or, as in the given example, the Law of Cosines. This law is especially handy when dealing with non-right triangles where typical 'SOHCAHTOA' rules for right triangles do not apply.

The strategies used to solve trigonometric problems often involve breaking down complex shapes into triangles, using known values to find unknowns, and applying algebraic manipulation, as seen in the step-by-step solution where we rearrange terms and potentially use the quadratic formula to solve for an unknown side length.

Exploring such solutions allows students to build a robust understanding of precalculus concepts and prepares them for the challenges of calculus.