Problem 12 - Trigonometry

Question

Each of the following problems refers to triangle \(A B C\). $$ \text { If } a=48 \text { inches, } b=84 \text { inches, and } C=120^{\circ} \text {, find } c \text {. } $$

Step-by-Step Solution

Answer

The length of side \(c\) is approximately 115.69 inches.

Step 1: Identify the Concept to be Used

This problem involves a triangle where two sides and the included angle are given. This suggests we can use the Law of Cosines to find the unknown side. The Law of Cosines is useful for non-right triangles.

Step 2: Recall the Law of Cosines

The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It is given by the formula: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]where \(a\), \(b\), and \(c\) are the sides of the triangle and \(C\) is the included angle.

Step 3: Substitute the Known Values

Substitute the given values \(a = 48\), \(b = 84\), and \(C = 120^{\circ}\) into the Law of Cosines formula: \[ c^2 = 48^2 + 84^2 - 2 \times 48 \times 84 \times \cos(120^{\circ}) \].

Step 4: Calculate \( \cos(120^{\circ}) \)

Recall that \( \cos(120^{\circ}) = -0.5\). This value is found using the cosine function for angles in standard position on the unit circle.

Step 5: Simplify the Equation

Substitute \(\cos(120^{\circ}) = -0.5\) into the equation: \[ c^2 = 48^2 + 84^2 + 2 \times 48 \times 84 \times 0.5 \].This simplifies to: \[ c^2 = 2304 + 7056 + 4032 \].

Step 6: Compute the Result

Add the numbers: \[ c^2 = 2304 + 7056 + 4032 = 13392 \].

Step 7: Find \(c\)

Take the square root of both sides to solve for \(c\): \[ c = \sqrt{13392} \approx 115.69 \].

Step 8: Final Step: Confirm Units

The sides of the triangle are given in inches, so the length of side \(c\) is also in inches.

Key Concepts

Law of Cosines

The Law of Cosines is an essential tool in trigonometry, especially when working with triangles that do not have a right angle. This mathematical formula helps us find the length of a side in a triangle when two sides and the angle between them (referred to as the included angle) are known.

The Law of Cosines is expressed as:

• \( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \)

Here, \( a \), \( b \), and \( c \) are the sides of the triangle, while \( C \) is the angle opposite to side \( c \). The utility of the Law of Cosines becomes most apparent in solving triangles where the Side-Angle-Side (SAS) condition is present.

This formula is especially useful for triangles that cannot be easily solved by the Pythagorean theorem, which only applies to right triangles. By accounting for the cosine of the included angle, the Law of Cosines provides a comprehensive approach to determine unknown sides in any triangle.

Triangle

A triangle, a fundamental shape in geometry, consists of three sides and three angles. Triangles are classified based on their angles and sides. For instance, there are **right triangles**, which contain a 90-degree angle, and **non-right triangles**, such as the given triangle in our problem.

The scenario in the exercise describes a common type of triangle problem where two sides and the included angle are known (Angle-Side-Side or ASS type may unfortunately also arise). This is a typical scenario where the Law of Cosines is particularly useful. In this specific case, the triangle given includes sides labeled \(a\), \(b\), and the angle \(C\) between them. Thus, it does not fit the 90-degree angle characteristic of right triangles.

So, for a non-right triangle like triangle ΔABC:

• The sides \( a \) and \( b \) form an angle \( C \).
• The goal is often to find the length of the side opposite one of the known angles or sides, such as side \( c \).

Understanding the properties and types of triangles helps in determining which trigonometric principles and formulas are best applied to solve a given problem.

Cosine Function

The cosine function, integral to trigonometry, relates the angles and sides of triangles, particularly through the Law of Cosines. In the context of our exercise, we handle the cosine of angle \( C = 120^{\circ} \).

In trigonometry, the cosine function is one of the primary trigonometric functions and is most often understood in terms of a right triangle as the ratio of the adjacent side to the hypotenuse. However, for angles beyond 90 degrees as in this problem, we employ the concept of the unit circle.

For our specific problem, \( \cos(120^{\circ}) \) was used:

• On the unit circle, any angle measured counterclockwise from the positive x-axis leads to finding values such as the cosine.
• The value of \( \cos(120^{\circ}) = -0.5 \), which is crucial in calculating the value of the unknown side \( c \) in our triangle.

This cosine value influences the calculation by adding terms in the Law of Cosines equation, which crucially affects finding the correct side length. Properly understanding and calculating the function's values are fundamental in solving trigonometric problems.