Problem 39 - Precalculus : Building Concepts and Connections

Question

A walkway around a flower bed in a park is made up of three straight sections that form the sides of a triangle. If the lengths of the sides are 26 feet, 24 feet, and 21 feet, what is the angle opposite the longest side?

Step-by-Step Solution

Answer

The angle opposite the longest side of the triangle is approximately 97.41 degrees.

Step 1: Identify the sides and the angle

Identify the longest side of the triangle which is 26 feet and denote it as 'c'. The remaining sides can be denoted as 'a' and 'b' which are 24 feet and 21 feet respectively. The aim is to find the angle opposite to the side 'c', let's denote it as θ.

Step 2: Apply the Law of Cosines

The Law of Cosines can be used to find the cosine of the angle θ. We do this by re-arranging the formula to look like this: \( \cos(θ) = (a^2 + b^2 - c^2) / (2ab) \). Substitute 'a', 'b' and 'c' with their respective values into the formula: \( \cos(θ) = (24^2 + 21^2 - 26^2) / (2*24*21) \). Calculate the result to get the cosine of θ.

Step 3: Calculate the angle

To get the angle θ, we use inverse cosine function. The result from the previous step was the cosine of the angle. We will pass this to the inverse cosine function, also known as arccosine, to get the angle in radians. Then, we can convert it to degrees by multiplying it by \( 180/\pi \) since 180 degrees is equivalent to \( \pi \) radians.

Key Concepts

Triangles

Triangles are fundamental geometric shapes composed of three edges and three vertices. The sum of its interior angles always equals 180 degrees. Understanding triangles is crucial in trigonometry, especially when solving problems that involve angle measurements and side lengths. There are several types of triangles based on their sides and angles:

• Scalene Triangle: All sides and angles are different.
• Isosceles Triangle: Has two equal sides and corresponding equal angles.
• Equilateral Triangle: All sides and angles are equal, with each angle being 60 degrees.

In the context of the given exercise, you are dealing with a scalene triangle because all sides are of different lengths: 26 feet, 24 feet, and 21 feet. Using properties of triangles and principles such as the Law of Cosines allows one to find unknown angles or sides, particularly in non-right angled triangles.

Inverse Trigonometric Functions

Inverse trigonometric functions are essential mathematical tools used to find angles when the values of some ratios are known. Inverse cosine, often denoted as arccos or \cos^{-1}, is one such function. It helps determine an angle from a given cosine value.When using the Law of Cosines to find the angle, you first calculate the cosine of the angle using the formula:\[\cos(\theta) = \frac{a^2 + b^2 - c^2}{2ab}\]After obtaining the cosine value, the inverse cosine is applied to find \(\theta\). By performing this calculation, you work out the exact measure of an angle from its trigonometric ratio.An important factor to remember is the domain of inverse cosine functions, which range from 0 to \(\pi\) radians in mathematical terms or 0 to 180 degrees in geometrical applications. The calculator often outputs the angle in radians, so converting it into degrees helps understand the measurement practically. The conversion uses the formula:\[\text{Angle in degrees} = \frac{180}{\pi} \times \text{Angle in radians}\]

Trigonometry

Trigonometry is the field of mathematics that studies the relationships between the sides and angles of triangles. It is particularly useful for solving problems involving triangles, most commonly using sine, cosine, and tangent functions.The Law of Cosines, like the one applied in this exercise, is crucial when dealing with non-right triangles. It is expressed as:\[c^2 = a^2 + b^2 - 2ab \cdot \cos\theta\]This equation expands our understanding of trigonometry beyond right triangles. It allows for finding unknown angles when all sides are known or calculating a side when two sides and their included angle are given.Trigonometry also employs the use of the unit circle, which represents angles and their respective sine and cosine values. Understanding the unit circle and various trigonometric identities is significant for comprehending how angle measures translate in real-world contexts.By mastering these concepts, you'll be better prepared to approach complex problems involving triangle geometries, much like solving the unknown angle in our original problem.