Problem 39 - Precalculus

Question

A tracking station has two telescopes that are 1.0 mile apart. The telescopes can lock onto a rocket after it is launched and record the angles of elevation to the rocket. If the angles of elevation from telescopes \(A\) and \(B\) are \(30^{\circ}\) and \(80^{\circ},\) respectively, then how far is the rocket from telescope \(A ?\)

Step-by-Step Solution

Answer

The rocket is approximately 1.05 miles from telescope A.

Step 1: Understand the Problem

The problem provides two telescopes 1 mile apart, and two angles of elevation from each telescope to a rocket. We need to find the distance from telescope \(A\) to the rocket.

Step 2: Visualize and Label the Triangle

Let's visualize the setup as a triangle where the telescopes form a base. Denote the distance from telescope \(A\) to the rocket as \(d\). We label this alongside a horizontal segment between \(A\) and \(B\), thereby creating triangle \(ABR\). Here, \(A\) and \(B\) are telescope positions, and \(R\) is the rocket.

Step 3: Utilize the Law of Sines

The Law of Sines can help relate the sides and angles of triangle \(ABR\). Given angles \(A = 30^{\circ}\), \(B = 80^{\circ}\), and \(C = 70^{\circ}\) (since sum of angles in a triangle is \(180^{\circ}\)), we use:\[\frac{d}{\sin(B)} = \frac{AC}{\sin(70^{\circ})}\] where \(AC = 1\, ext{mile}\).

Step 4: Calculate the Distance Using the Law of Sines

Substitute known values into the equation from the Law of Sines:\[\frac{d}{\sin(80^{\circ})} = \frac{1}{\sin(70^{\circ})}\]Solve for \(d\):\[d = \frac{\sin(80^{\circ}) \times 1}{\sin(70^{\circ})}\] Calculating using a calculator:\(d \approx \frac{0.985}{0.94} \approx 1.0489\). So, the distance \(d\) is approximately 1.05 miles.

Key Concepts

Triangles

Triangles, as basic geometric figures, compose of three sides and three angles. In the context of the Law of Sines, triangles are central in solving complex problems related to distance and angles. Visualizing a problem as a triangle helps in breaking down the problem into manageable parts.
One key aspect to remember about triangles is that the sum of their interior angles is always 180 degrees. This property is crucial when using trigonometric relationships because:

• It helps identify unknown angles when at least two angles are known.
• Ensures that calculations remain consistent whether dealing with right or oblique triangles.
• Allows the application of the Law of Sines, which requires knowledge of the angles and sides of triangles.

Using these properties effectively can provide the distance and positional information needed in a variety of scenarios, like calculating the path of a rocket from vantage points on Earth.

Angles of Elevation

Angles of elevation are essential when it comes to solving real-world trigonometry problems involving height and distance. Imagine standing on the ground and looking up at an object like a rocket. The angle you make with the horizontal line to the line of sight of the object is the angle of elevation.
These angles are pivotal in solving problems with the Law of Sines since:

• They provide a means to relate the vertical height of objects with horizontal distance.
• The angles help in forming triangles, which further substantiate trigonometric calculations.
• Understanding these angles can aid in calculating distances indirectly using trigonometric identities.

When such angles are obtained from different positions, they offer different perspectives which can be utilized in triangulation methods to pinpoint the exact distances and positions of the target objects such as rockets.

Trigonometry

Trigonometry deals with the study of triangles, specifically the relationships between their angles and sides. It extends beyond simple geometric ideas and provides tools for calculating unknown lengths and angles even in difficult-to-measure scenarios.
A useful tool in trigonometry is the Law of Sines. It states that in any triangle:\[\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\]where \(a\), \(b\), and \(c\) are the sides opposite the angles \(A\), \(B\), and \(C\) respectively. This law proves invaluable when dealing with oblique triangles—triangles that do not contain a right angle.

• It helps in finding unknown sides when two angles and one side are known.
• Also, it assists in finding unknown angles if two sides and one non-enclosed angle are known.
• Illustrates how angles and sides correlate in a triangle formed by line-of-sight scenarios.

With the Law of Sines, you can solve our original exercise without direct measurements, using only the angles of elevation and a known distance, making it a powerful method in trigonometric applications.