Problem 47 - Precalculus Mathematics for Calculus

Question

Field A triangular field has sides of lengths \(22,36,\) and 44 yd. Find the largest angle.

Step-by-Step Solution

Answer

The largest angle is approximately 97.69°.

Step 1: Identify the longest side

In a triangle, the largest angle is opposite the longest side. For this triangle with sides 22 yd, 36 yd, and 44 yd, the longest side is 44 yd.

Step 2: Use the Cosine Rule

To find an angle in a triangle when all side lengths are known, use the cosine rule: \[ c^2 = a^2 + b^2 - 2ab \cos(C) \]Here, \(c\) is the longest side, 44 yd, and \(a\) and \(b\) are the other two sides, 22 yd and 36 yd. Substitute these values into the formula.

Step 3: Substitute known values

Substitute \(c = 44\), \(a = 22\), and \(b = 36\) into the cosine rule equation:\[ 44^2 = 22^2 + 36^2 - 2 \times 22 \times 36 \cos(C) \]

Step 4: Simplify the equation

Calculate each term: \[ 1936 = 484 + 1296 - 1584 \cos(C) \]Further simplify:\[ 1936 = 1780 - 1584 \cos(C) \]

Step 5: Solve for \(\cos(C)\)

Rearrange the equation to solve for \(\cos(C)\):\[ 1584 \cos(C) = 1780 - 1936 \]\[ \cos(C) = \frac{-156}{1584} \]\[ \cos(C) = -\frac{13}{132} \]

Step 6: Calculate angle C

Use an inverse cosine function to find angle \(C\): \[ C = \cos^{-1}\left(-\frac{13}{132}\right) \]Calculate to find the angle in degrees.

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that delves into the relationships between the angles and sides of triangles. It's particularly useful for solving problems where you need to find unknown angles or lengths in triangles. When all sides of a triangle are known, like in this exercise, the cosine rule is a powerful tool. It provides a direct way to calculate an unknown angle without needing any angle-related initial information.

• Principles of trigonometry are fundamental in both algebraic calculations and geometric visualizations.
• It's not only about triangles – trigonometry also extends to understanding wave patterns and oscillations.
• Cosine, sine, and tangent are the basic trigonometric functions that help in angle assessments.

Understanding trigonometry allows for precise calculations in fields such as engineering, physics, and even computer graphics. The cosine rule, in particular, is exceptionally handy when no angle measures are initially available.

Triangle Geometry

Understanding the geometry of triangles is crucial in many areas of math. Triangles are geometric shapes made up of three sides and three angles. The properties of these elements define different types of triangles, such as

• equilateral – all sides and angles are equal,
• isosceles – two sides and angles are equal,
• scalene – all sides and angles are different.

In this exercise, we are dealing with a scalene triangle. These types have no sides or angles of equal measure, requiring specific strategies like the cosine rule to solve problems related to them. Recognizing that the largest angle is opposite the longest side is a key principle, which guides you to finding the solution efficiently. This connection between side lengths and angles is what makes the geometry of triangles both fascinating and practical.

Angle Calculation

Calculating angles in a triangle is a task that often involves using trigonometric identities and principles, such as the cosine rule. When you know all three sides of a triangle, the cosine rule becomes a straightforward method to find any angle. This rule is formally stated as \[ c^2 = a^2 + b^2 - 2ab \cos(C) \]where \(c\) is the side opposite the angle \(C\). By rearranging this formula, you can find the cosine of angle \(C\), and then use a calculator to find the angle itself by applying the inverse cosine function.

• The angle calculations are often expressed in degrees.
• Using a calculator is essential for inverse trigonometric functions.
• Always ensure that your calculator is set to the correct mode for angle calculations, usually degrees in these instances.

These steps allow you to find precise measurements, ensuring your answers are accurate and reliable. Angle calculations form the foundation for deeper explorations into more complex trigonometric concepts.