Problem 60 - Thomas Calculus

Question

A triangle has sides \(a=2\) and \(b=3\) and angle \(C=40^{\circ} .\) Find the length of side \(c .\)

Step-by-Step Solution

Answer

The length of side \(c\) is approximately \(1.951\).

Step 1: Identify Known Elements

We know the lengths of two sides of the triangle, which are \(a = 2\) and \(b = 3\), and the included angle \(C = 40^{\circ}\). We need to find the length of side \(c\).

Step 2: Apply the Law of Cosines

The Law of Cosines states that for any triangle with sides \(a\), \(b\), and \(c\), and an angle \(C\) opposite side \(c\), the equation is \(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\). Substitute the known values: \(c^2 = 2^2 + 3^2 - 2 \cdot 2 \cdot 3 \cdot \cos(40^{\circ})\).

Step 3: Simplify the Expression

Calculate \(2^2 = 4\) and \(3^2 = 9\), then substitute back: \(c^2 = 4 + 9 - 2 \cdot 2 \cdot 3 \cdot \cos(40^{\circ})\). Simplify to \(c^2 = 13 - 12 \cdot \cos(40^{\circ})\).

Step 4: Calculate \(\cos(40^{\circ})\)

Use a calculator to find \(\cos(40^{\circ})\approx 0.7660\).

Step 5: Complete the Substitution and Calculate

Substitute \(\cos(40^{\circ})\) into the equation: \(c^2 = 13 - 12 \cdot 0.7660\). Calculate \(12 \cdot 0.7660 = 9.192\). Then \(c^2 = 13 - 9.192 = 3.808\).

Step 6: Find the Length of Side \(c\)

Take the square root of \(3.808\) to find \(c\), which is approximately \(c \approx \sqrt{3.808} \approx 1.951\). Thus, the length of side \(c\) is approximately \(1.951\).

Key Concepts

Trigonometry

Trigonometry is an essential branch of mathematics that deals with the study of triangles, particularly right-angled triangles. It involves the relationships between the sides and angles of triangles and uses functions such as sine, cosine, and tangent. In dealing with non-right triangles, like the one in our given problem, we rely on the Law of Cosines to find unknown sides or angles.

Using trigonometry allows us to calculate dimensions that are otherwise challenging to measure directly. In this exercise, trigonometry becomes pivotal, as it facilitates the calculation of the length of side \(c\) of a triangle when knowing two sides and the included angle using the Law of Cosines.

These concepts are not just limited to theoretical exercises but are applicable in real-world scenarios like navigation, construction, and physics. Grasping trigonometry helps us understand the geometrical arrangement of objects around us and equips us with tools to solve practical problems.

Triangle Geometry

Triangle geometry involves understanding the property and relationships between the sides and angles of triangles. A triangle is a three-sided polygon, and its internal angles add up to \(180^\circ\). When analyzing triangles, certain formulas and theorems help solve for unknown dimensions or angles.

In triangle geometry, the Law of Cosines is particularly useful for solving triangles when the triangle does not have a right angle. This formula, which relates the lengths of the sides of a triangle to the cosine of one of its angles, is a direct extension of the Pythagorean theorem, useful when dealing with oblique triangles. In our problem, it helps calculate the missing side, \(c\), with the formula: \[c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\]

This opens up possibilities for calculating not only side lengths but also angles, making it a versatile tool in geometrical analysis and solving.

Angle Measurement

Angle measurement plays a critical role in triangle problems, as angles determine how the sides are oriented relative to each other. Angles are measured in degrees or radians, with \(360\degree\) making a full circle and \(180\degree\) for a straight angle. In the context of triangles, understanding how to measure and use angles is crucial.

In our exercise, angle \(C\) is given as \(40^\circ\), an included angle between sides \(a\) and \(b\). This information, combined with the Law of Cosines, allows us to solve for the unknown side \(c\). The use of cosine predominantly in this formula showcases how angle measurements directly influence the calculation of side lengths in triangles, highlighting their importance in the process.

By harnessing the properties of angles, we can delve deeper into understanding different shapes and patterns in geometry, paving the way for accurate and efficient problem-solving. Whether using a calculator to find trigonometric values like \(\cos(40^\circ)\), or manually estimating angles, proficiency in angle measurements enhances our capability to tackle various mathematical challenges.