Problem 25 - Trigonometry

Question

Distance A 51 -foot wire running from the top of a tent pole to the ground makes an angle of \(58^{\circ}\) with the ground. If the length of the tent pole is 44 feet, how far is it from the bottom of the tent pole to the point where the wire is fastened to the ground? (The tent pole is not necessarily perpendicular to the ground.)

Step-by-Step Solution

Answer

The horizontal distance is approximately 46.46 feet.

Step 1: Understand the Problem

We have a wire making an angle of \(58^{\circ}\) with the ground and is 51 feet long. The wire is attached to the ground, forming a triangle with the tent pole, which is 44 feet long. We need to find the horizontal distance from the bottom of the tent pole to the point where the wire touches the ground.

Step 2: Identify the Right Triangle

The problem forms a triangle with the tent pole, the wire, and the ground. The wire acts as the hypotenuse with the ground, and the horizontal distance we are looking for is the base. Since the wire connects both ends, the triangle's third side is the tent pole.

Step 3: Apply the Law of Cosines

Since it's not mentioned that the triangle is a right triangle, use the Law of Cosines to find the horizontal distance. This states that in a triangle \(c^2 = a^2 + b^2 - 2ab\cos(C)\), where \(c\) is the side opposite angle \(C\). Here, \(c = 44\), \(a = 51\), angle \(C = 58^{\circ}\), and we need to find \(b\) (the horizontal distance): \[b^2 = 51^2 + 44^2 - 2 \cdot 51 \cdot 44 \cdot \cos(58^{\circ})\]

Step 4: Calculations

Now, compute each part of the equation:- \(51^2 = 2601\)- \(44^2 = 1936\)- \(2 \cdot 51 \cdot 44 = 4488\)- \(\cos(58^{\circ}) \approx 0.5299\)Calculate the expression:\[b^2 = 2601 + 1936 - 4488 \cdot 0.5299\]\[b^2 = 4537 - 2377.3512\]\[b^2 = 2159.6488\]Then, take the square root:\[b \approx \sqrt{2159.6488} \approx 46.46\]

Step 5: Conclusion

The horizontal distance from the bottom of the tent pole to the point where the wire is fastened to the ground is approximately 46.46 feet.

Key Concepts

Trigonometry

Trigonometry is the branch of mathematics that deals with the relationships between angles and sides in triangles. One of the most powerful tools in trigonometry is the use of right triangles to find indirect measurements. However, when a triangle is not right-angled, we rely on relationships like the Law of Cosines and the Law of Sines to solve problems.

The Law of Cosines is particularly useful when you know:

• Two sides of a triangle and the angle between them.
• Or, all three sides of a triangle and want to find an angle.

In this exercise, the wire running from a tent to the ground forms a triangle with the ground and the pole, where the angle between the wire and the ground is given. This setup allows us to apply the Law of Cosines to find the exact length of the side, which is not perpendicular, showing the true power of trigonometry in solving real-world problems.

Triangle Geometry

Triangles are one of the fundamental shapes in geometry, consisting of three sides and three angles. In the context of solving problems involving triangles, knowing whether a triangle is right, acute, or obtuse is crucial. For this particular problem, the angle made by the wire with the ground is 58 degrees, which classifies the triangle as an acute triangle.

In any triangle, the sum of angles always adds up to 180 degrees, helping find missing angles when given enough information about the other angles. This constraint ensures that every side and angle relate to one another in specific ways, forming the triangle's geometry.

The Law of Cosines fits perfectly into triangle geometry when the triangle doesn't include a 90-degree angle. It effectively extends the applicability of right triangle principles to any triangle, allowing us to compute unknown side lengths or angles when we have partial information.

Angle Measurement

Angle measurement is a vital part of understanding and solving geometric problems. Angles are typically measured in degrees, with the full circle comprising 360 degrees. For triangles, understanding the relationships between angles is essential for determining unknown features of the shape.

When working with angles, it's important to distinguish between acute (less than 90 degrees), right (exactly 90 degrees), and obtuse angles (greater than 90 degrees). In this exercise, we have a triangle featuring an acute angle of 58 degrees.

Accurate angle measurement ensures precise calculations, especially when using trigonometric functions like cosine, which requires converting angles to radians when calculated in many mathematical software. The knowledge of the angle's measure allows us to correctly apply the Law of Cosines: knowing the angle helps us understand the spatial relationship between the hypotenuse and the base, aiding in the accurate computation of the unknown side.