Problem 59 - Thomas Calculus

Question

A triangle has sides \(a=2\) and \(b=3\) and angle \(C=60^{\circ} .\) Find the length of side \(c .\)

Step-by-Step Solution

Answer

The length of side \(c\) is \(\sqrt{7}\).

Step 1: Understanding the problem

We are given a triangle with two sides, \(a = 2\) and \(b = 3\), and an angle \(C = 60^\circ\) between them. We need to find the length of the third side \(c\).

Step 2: Use the Law of Cosines

To find the length of side \(c\), we use the Law of Cosines formula: \[c^2 = a^2 + b^2 - 2ab\cos(C)\.\] This formula relates the lengths of the sides of a triangle to the cosine of one of its angles.

Step 3: Substitute the known values

We substitute the given values into the Law of Cosines formula: \[c^2 = 2^2 + 3^2 - 2 \times 2 \times 3 \times \cos(60^\circ)\.\] Simplify each part: \(2^2 = 4\), \(3^2 = 9\), and \(\cos(60^\circ) = 0.5\).

Step 4: Calculate the expression

Continue simplifying:\[c^2 = 4 + 9 - 12 \times 0.5\.\] This simplifies to:\[c^2 = 13 - 6 = 7\.\]

Step 5: Solve for \(c\)

Since we have \(c^2 = 7\), take the square root of both sides to find \(c\):\[c = \sqrt{7}\.\]

Step 6: Provide the final solution

The length of side \(c\) is \(\sqrt{7}\).

Key Concepts

Understanding Trigonometry

Trigonometry is a branch of mathematics focused on the study of triangles, particularly right triangles and the relationships between their angles and sides. This study is crucial for various applications ranging from simple geometry problems to advanced physics and engineering. It provides tools to measure distances and angles, making it indispensable in fields like astronomy and navigation.

Key elements of trigonometry include functions such as sine, cosine, and tangent. These functions relate the angles of a triangle to the lengths of its sides. When dealing with non-right triangles, like in our exercise, we often use the Sine Rule and the Cosine Rule (Law of Cosines) to solve problems.

The Law of Cosines, in particular, is a powerful tool in trigonometry for finding unknown sides or angles in a triangle where other trigonometric methods may not apply. Understanding these concepts is essential for mastering the problem-solving techniques seen in trigonometry.

Basics of Triangle Geometry

Triangle geometry is all about understanding the different types and properties of triangles. A triangle is a three-sided polygon and is the simplest form of a polygon. Each triangle consists of three sides and three angles. Importantly, the sum of the internal angles is always 180 degrees.

There are several types of triangles, categorized based on their sides or angles, such as:

• Right Triangle: One angle is 90 degrees.
• Equilateral Triangle: All sides and angles are equal.
• Isosceles Triangle: Two sides (and angles) are equal.
• Scalene Triangle: All sides and angles are different.

In our exercise, we are working with an oblique triangle, meaning that none of its angles are right angles.

Understanding these properties helps to determine how to apply geometric formulas, such as the Law of Cosines, to find missing sides or angles. Each type of triangle has unique properties that dictate how we calculate various elements within it.

Exploring the Cosine Function

The cosine function is one of the primary trigonometric functions used in mathematics. It is critically important in the study of triangle geometry, especially when applying the Law of Cosines.

The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the hypotenuse. In the context of non-right triangles, this function helps solve problems involving unknown sides or angles.

The mathematical expression for cosine, in terms of a unit circle, is:

• If you have a circle with a radius of 1 unit centered at the origin, the cosine of an angle is the x-coordinate of the point where the terminal side of the angle intersects the circle.

In practical applications, especially in our exercise, the cosine function allows us to incorporate known angles to find unknown side lengths, enhancing our ability to solve triangle problems where not all angles or sides are initially known.