Problem 38 - Algebra and Trigonometry

Question

Radio Antenna A short-wave radio antenna is supported by two guy wires, 165 \(\mathrm{ft}\) and 180 \(\mathrm{ft}\) long. Each wire is attached to the top of the antenna and anchored to the ground, at two anchor points on opposite sides of the antenna. The shorter wire makes an angle of \(67^{\circ}\) with the ground. How far apart are the anchor points?

Step-by-Step Solution

Answer

The distance between the anchor points is approximately 190.72 ft.

Step 1: Analyze the Problem

Visualize the problem as a triangle formed by the two anchor points on the ground and the top of the antenna. You have a triangle with sides 165 ft, 180 ft, and the distance between the anchor points, which is the side we need to find. The angle between the ground and the shorter wire is given as \(67^{\circ}\). This is a non-right triangle problem and can be solved using the law of cosines.

Step 2: Apply the Law of Cosines

To find the distance between the anchor points, use the law of cosines. The law of cosines states: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]Here, \(a = 165\) ft, \(b = 180\) ft, and \(C = 67^{\circ}\). Substitute these values into the formula to find \(c\).

Step 3: Calculate Using the Formula

Substitute the known values into the equation:\[ c^2 = 165^2 + 180^2 - 2 \cdot 165 \cdot 180 \cdot \cos(67^{\circ}) \]First, calculate each term separately:- \(165^2 = 27225\)- \(180^2 = 32400\)- Calculate \(2 \cdot 165 \cdot 180 = 59400\)- Calculate \(\cos(67^{\circ})\) using a calculator.Then, compute the entire expression.

Step 4: Solve for Distance

Calculate \(c^2\):\[ c^2 = 27225 + 32400 - 59400 \cdot \cos(67^{\circ}) \]Using a calculator, find \(\cos(67^{\circ}) \approx 0.3907\) and compute:\[ c^2 = 27225 + 32400 - 59400 \times 0.3907 \]\[ c^2 = 27225 + 32400 - 23217.58 \]\[ c^2 = 36407.42 \]Take the square root to find \(c\):\[ c \approx \sqrt{36407.42} \approx 190.72 \text{ ft} \]

Step 5: Present the Result

The distance between the two anchor points, calculated using the law of cosines, is approximately 190.72 ft.

Key Concepts

Triangles

In geometry, a triangle is a three-sided polygon that consists of three edges and three vertices. Every triangle has three angles whose sum always equals 180 degrees. Triangles can be classified into various types based on their side lengths or their angle measures.

• Based on sides: Equilateral (all sides equal), Isosceles (two sides equal), Scalene (no sides equal).
• Based on angles: Acute (all angles less than 90 degrees), Right (one angle is 90 degrees), Obtuse (one angle more than 90 degrees).

Understanding these properties is essential when dealing with triangle-related problems, like the one with our antenna triangle. Knowing the types helps determine the best approach to solve the problem. For non-right triangles, where there's no 90-degree angle, we rely on laws like the law of sines or law of cosines to find missing sides and angles.

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between angles and sides of triangles. It is especially useful in solving real-world problems involving non-right triangles, such as the one described in the antenna problem. Trigonometry functions include sine, cosine, and tangent, which relate the angles of a triangle to the lengths of its sides.One of the main tools from trigonometry used here is the law of cosines. This law allows us to calculate one side of a triangle when we know the other two sides and the angle between them.The formula given by the law of cosines is:\[ c^2 = a^2 + b^2 - 2ab \, \cos(C) \] This formula expands the Pythagorean theorem, which is limited to right triangles, making it applicable to any type of triangle. Calculating \( \cos(C) \) helps in finding the desired side when dealing with non-right triangles.

Non-right Triangles

Unlike right triangles, which have one angle of 90 degrees, non-right triangles do not have a right angle. These triangles require different methods like the law of cosines or the law of sines to solve, as they can't be solved using simple right-angle trigonometry. In our exercise, we were faced with a scalene, non-right triangle problem. To find the distance between the anchor points of the antenna, we applied the law of cosines. This method is useful when: - You know two sides of a triangle and the angle between them (SAS situation). - You know all three sides and want to find an angle (SSS situation). Using these laws, we can tackle complex geometric problems involving non-right triangles, finding missing lengths and angles with accuracy and ease.