Problem 70 - Precalculus

Question

In an oblique triangle \(A B C, \beta=\frac{2 \pi}{9}, \gamma=\frac{5 \pi}{9},\) and \(a=200 \mathrm{ft} .\) Find the length of \(c .\) Round your answer to the nearest unit.

Step-by-Step Solution

Answer

The length of \(c\) is approximately 307 ft.

Step 1: Understand the Given Triangle and Angles

We are given an oblique triangle \(ABC\) with angles \(\beta\) and \(\gamma\) provided as \(\beta=\frac{2\pi}{9}\) and \(\gamma=\frac{5\pi}{9}\). Since the sum of angles in a triangle is \(\pi\), calculate \(\alpha\) as \(\alpha = \pi - \beta - \gamma\).

Step 2: Calculate Angle \(\alpha\)

Substitute the given values of \(\beta\) and \(\gamma\) into the equation: \[\alpha = \pi - \frac{2\pi}{9} - \frac{5\pi}{9} = \pi - \frac{7\pi}{9} = \frac{2\pi}{9}\]. So, \(\alpha = \frac{2\pi}{9}\).

Step 3: Apply the Law of Sines

Use the Law of Sines: \(\frac{a}{\sin(\alpha)} = \frac{c}{\sin(\gamma)}\). We have \(a = 200\) ft, \(\alpha = \frac{2\pi}{9}\), and \(\gamma = \frac{5\pi}{9}\).

Step 4: Substitute Known Values into the Law of Sines

By substituting the values, we get: \[\frac{200}{\sin(\frac{2\pi}{9})} = \frac{c}{\sin(\frac{5\pi}{9})}\].

Step 5: Solve for \(c\)

Rearrange to solve for \(c\): \[c = \frac{200 \cdot \sin(\frac{5\pi}{9})}{\sin(\frac{2\pi}{9})}\]. Calculate the sine values and then compute \(c\).

Step 6: Compute the Sine Values

Calculate \(\sin(\frac{2\pi}{9})\) and \(\sin(\frac{5\pi}{9})\) using a calculator. Approximate results are \(\sin(\frac{2\pi}{9}) \approx 0.6428\) and \(\sin(\frac{5\pi}{9}) \approx 0.9848\).

Step 7: Calculate \(c\) and Round the Answer

Substitute the sine values to find \(c\): \[c \approx \frac{200 \cdot 0.9848}{0.6428} \approx 306.67\]. Round to the nearest unit: \(c \approx 307\) ft.

Key Concepts

Oblique Triangle

An oblique triangle is any triangle that does not have a right angle. This means that all of its internal angles are less than 90 degrees or greater than 90 degrees. In contrast to right triangles, oblique triangles can be either acute or obtuse. In the given exercise, triangle \(ABC\) is oblique because none of its angles equal \( \frac{\pi}{2} \) radians, which is equivalent to 90 degrees.

Understanding the nature of an oblique triangle is essential when solving for unknown sides or angles because it determines what mathematical tools and formulas we use. Unlike right triangles, where the Pythagorean theorem applies, oblique triangles require methods like the Law of Sines or the Law of Cosines for calculations, making angle calculations and trigonometric functions very important in solving these problems.

In this exercise, the angles \( \beta \) and \( \gamma \) are provided, and we found \( \alpha \) by knowing the sum of the angles in a triangle is always \( \pi \) radians. This step sets the stage for applying the Law of Sines efficiently.

Angle Calculation

Calculating angles in a triangle involves understanding that the sum of internal angles is always a constant. In a triangle, this sum is equivalent to \( \pi \) radians (or 180 degrees). Given two angles of an oblique triangle, the third angle \( \alpha \) can be easily determined. The key formula for this is:

• \[ \alpha = \pi - \beta - \gamma \]

By substituting the given angles \( \beta = \frac{2\pi}{9} \) and \( \gamma = \frac{5\pi}{9} \), we can find \( \alpha \) without any ambiguity. The calculation is straightforward and crucial because knowing all three angles enables the use of the Law of Sines.

This foundational understanding paves the way for further operations using trigonometric identities and helps maintain a consistent calculation approach throughout the process of solving the triangle.

Trigonometric Functions

Trigonometric functions play a pivotal role in solving oblique triangles, as seen in this exercise. Here, the trigonometric function of the sine is particularly important due to its appearance in the Law of Sines. The Law of Sines relates the sides of a triangle to the sines of its angles, and its formula is:

• \[ \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)} \]

In the exercise, we needed to find the unknown side \(c\). By using the information that \( a = 200 \) ft, \( \alpha = \frac{2\pi}{9} \), and \( \gamma = \frac{5\pi}{9} \), substituting these values into the Law of Sines allowed us to solve for \( c \).

Calculating the sine of an angle can be done using a calculator, which yields approximate decimals essential for accurate calculations. For instance, \( \sin(\frac{2\pi}{9}) \approx 0.6428 \) and \( \sin(\frac{5\pi}{9}) \approx 0.9848 \). Filling in these values led us to calculate \( c \).

Overall, understanding how to apply trigonometric functions efficiently is crucial for solving oblique triangles, as they directly connect angle measures to side lengths.