Problem 61 - Thomas Calculus

Question

The law of sines Thelaw of sines says that if \(a, b,\) and \(c\) are the sides opposite the angles \(A, B,\) and \(C\) in a triangle, then $$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$$ Use the accompanying figures and the identity \(\sin (\pi-\theta)=\) \(\sin \theta,\) if required, to derive the law.

Step-by-Step Solution

Answer

The Law of Sines states \(\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\) based on equating triangle area expressions using sine.

Step 1: Understand the Triangle Setup

Recall that the Law of Sines applies to any triangle, not just right triangles. For a triangle with sides labeled as \(a\), \(b\), and \(c\), opposite to angles \(A\), \(B\), and \(C\) respectively, we seek to establish a relationship between the sides and their opposing angles.

Step 2: Set Up the Sine Ratios

Consider a triangle \(ABC\). Recall that across any triangle, for the angle \(A\), we can find the opposite side \(a\) and similarly for \(B\) and \(C\). The primary aim is to establish that the following ratio equivalence holds: \(\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\).

Step 3: Use the Sine Definition on Area

Consider the area \(K\) of the triangle, which can be written using sine: \(K = \frac{1}{2}ab\sin C = \frac{1}{2}bc\sin A = \frac{1}{2}ca\sin B\). These relationships come from the definition of the area of a triangle (\(\text{Area} = \frac{1}{2}ab\sin C\), etc.).

Step 4: Equate the Area Expressions

Since the area can be expressed in three different forms involving the sines of angles \(A\), \(B\), and \(C\), we equate: \(\frac{1}{2}ab\sin C = \frac{1}{2}bc\sin A = \frac{1}{2}ca\sin B\).

Step 5: Solve for the Sines of Angles

From equated area equations, isolate \(\sin C\) on one side: \(\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\) by dividing each term by the corresponding side length, proving the Law of Sines.

Key Concepts

Trigonometry

Trigonometry is the branch of mathematics that explores the relationships between the angles and sides of triangles. At its core, trigonometry functions using the concepts of sine, cosine, and tangent. These functions help in calculating various properties of a triangle. The Law of Sines is a fundamental theorem in trigonometry that aids in solving triangles. It is especially useful when you have a non-right triangle where traditional methods fall short.

• Sine, cosine, and tangent are the primary trigonometric functions used for calculations.
• These functions help relate angles to their opposite sides in a triangle.
• Trigonometry applies to real-world problems like navigation, architecture, and physics.

Understanding trigonometry opens doors to solving complex problems involving triangles effortlessly. It acts as a toolkit for discovering unknown angles or side lengths by using given or assumed values.

Triangle Properties

Triangles are basic geometric figures, composed of three sides and three angles. Each triangle has specific properties that distinguish it from other shapes. These properties are crucial when dealing with problems involving the Law of Sines. The primary properties include:

• The sum of internal angles in any triangle is always 180°.
• Triangles can be classified based on their angles (such as acute, obtuse) or their sides (like scalene, isosceles, and equilateral).
• The longest side is opposite the largest angle in any triangle.
• In a scalene triangle, all sides and angles are different.
• Isosceles triangles have at least two equal sides and angles.

Recognizing these properties allows for the correct application of the Law of Sines and other trigonometric principles.

Sine Function

The sine function is one of the most important aspects of trigonometry. It relates an angle of a triangle to the ratio of the length of the side opposite the angle and the triangle's hypotenuse. In any triangle, the sine of an angle is crucial for deriving relationships between angles and sides. The sine identity, \( \sin(\pi - \theta) = \sin \theta \), plays a vital role in establishing that two supplementary angles have equal sine values.

• The sine function is defined as \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \).
• It is usually used when dealing with right-angle triangles but can extend to any triangle as proven by the Law of Sines.
• The amplitude of the sine function is its maximum extent, which is 1.

This fundamental relationship provided by the sine function forms the basis for understanding how angles and sides interconnect across different kinds of triangles.

Geometry

Geometry, the study of shapes, sizes, and the properties of space, forms the groundwork for understanding triangles and their properties. In geometry, different rules and theorems apply to how shapes are formed and how they relate to each other. Geometry allows us to visually interpret and analyze figures enhancing our comprehension of mathematical principles.

• Basic geometric concepts include points, lines, angles, and surfaces.
• It builds the framework for understanding complex structures by using simpler components.
• The application of geometry can be seen in art, engineering, and nature.

The Law of Sines is a geometric principle that connects the dots between different properties of a triangle, making complicated calculations simpler and comprehensible.