Law of Cosines and its Applications

Introduction

Key Concepts

Understanding 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 particularly useful in determining unknown sides or angles in non-right-angled triangles. The formula is given by:

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

where:

• a, b, and c are the lengths of the sides of the triangle.
• C is the angle opposite side c.

This formula generalizes the Pythagorean theorem, which is a specific case of the Law of Cosines when angle C is 90 degrees. In such a scenario, $\cos(90^\circ) = 0$, and the Law of Cosines simplifies to:

$$c^2 = a^2 + b^2$$

Derivation of the Law of Cosines

The Law of Cosines can be derived using geometric principles and the Pythagorean theorem. Consider a triangle with sides a, b, and c, and angle C opposite side c. By dropping a perpendicular from the vertex of angle C to the base formed by side c, we can create two right-angled triangles. Let the length of the perpendicular be h, and the segments of the base be d and c - d.

Applying the Pythagorean theorem to both right-angled triangles:

$$a^2 = h^2 + d^2$$ $$b^2 = h^2 + (c - d)^2$$

Subtracting these equations gives:

$$a^2 - b^2 = d^2 - (c - d)^2$$ $$a^2 - b^2 = d^2 - c^2 + 2cd - d^2$$ $$a^2 - b^2 = -c^2 + 2cd$$ $$2cd = a^2 - b^2 + c^2$$ $$d = \frac{a^2 - b^2 + c^2}{2c}$$

Now, using the definition of cosine in the original triangle:

$$\cos(C) = \frac{d}{a}$$

Substituting d from above:

$$\cos(C) = \frac{a^2 - b^2 + c^2}{2ac}$$

Rearranging terms leads to the Law of Cosines:

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

Applications of the Law of Cosines

The Law of Cosines is instrumental in various scenarios, including:

• Solving Triangles: Determining unknown sides or angles in any triangle, especially when the triangle is not right-angled.
• Navigation and Surveying: Calculating distances and angles in land surveying and navigation tasks.
• Physics and Engineering: Analyzing force vectors, structural integrity, and motion dynamics.
• Computer Graphics: Computing distances and angles for rendering 3D models.

Solving Triangles Using the Law of Cosines

To solve a triangle means to find all its side lengths and angles given certain known elements. The Law of Cosines is particularly useful in two scenarios:

1. SAS (Side-Angle-Side): Given two sides and the included angle, the Law of Cosines can find the third side.
2. SSS (Side-Side-Side): Given all three sides, the Law of Cosines can determine each angle of the triangle.

SAS Example:

Given:

• Side a = 7 units
• Side b = 10 units
• Angle C = 45 degrees

Find side c:

$$c^2 = a^2 + b^2 - 2ab\cos(C)$$ $$c^2 = 7^2 + 10^2 - 2 \times 7 \times 10 \times \cos(45^\circ)$$ $$c^2 = 49 + 100 - 140 \times \frac{\sqrt{2}}{2}$$ $$c^2 = 149 - 70\sqrt{2}$$ $$c \approx \sqrt{149 - 70 \times 1.414}$$ $$c \approx \sqrt{149 - 98.98}$$ $$c \approx \sqrt{50.02}$$ $$c \approx 7.07 \text{ units}$$

SSS Example:

Given:

• Side a = 8 units
• Side b = 15 units
• Side c = 17 units

Find angle C:

$$c^2 = a^2 + b^2 - 2ab\cos(C)$$ $$289 = 64 + 225 - 2 \times 8 \times 15 \times \cos(C)$$ $$289 = 289 - 240\cos(C)$$ $$0 = -240\cos(C)$$ $$\cos(C) = 0$$ $$C = 90^\circ$$

Real-World Example: Navigation

Imagine a navigator needs to determine the distance between two points, A and B, given their respective distances and angles from a third point, C. By applying the Law of Cosines, the navigator can accurately compute the direct distance between points A and B, which is essential for charting precise courses.

Advanced Concepts

Theoretical Extensions of the Law of Cosines

The Law of Cosines can be extended to spherical and vector contexts, providing a bridge between planar and three-dimensional analyses. In spherical trigonometry, for instance, the Law of Cosines adapts to account for the curvature of the sphere:

$$\cos(a) = \cos(b)\cos(c) + \sin(b)\sin(c)\cos(A)$$

where a, b, and c are sides of a spherical triangle, measured as angles from the sphere's center, and A is the angle opposite side a.

This extension is crucial in fields like astronomy and geodesy, where calculations often involve celestial spheres or the Earth's curvature.

Vector Applications of the Law of Cosines

In vector mathematics, the Law of Cosines facilitates the computation of the angle between two vectors or the magnitude of their resultant. Given two vectors u and v, the magnitude of their sum w = u + v can be found using the Law of Cosines:

$$||w||^2 = ||u||^2 + ||v||^2 + 2||u|| \cdot ||v|| \cdot \cos(\theta)$$

where θ is the angle between vectors u and v.

This principle is widely applied in physics to resolve forces, velocities, and other vector quantities.

Proof of the Law of Cosines Using Coordinate Geometry

An alternative proof of the Law of Cosines employs coordinate geometry. Consider triangle ABC with vertex A at the origin (0,0), vertex B at (c,0), and vertex C at (d, e). Using the distance formula, the lengths of sides AB, AC, and BC can be expressed as:

$$AB = c$$ $$AC = \sqrt{d^2 + e^2}$$ $$BC = \sqrt{(c - d)^2 + e^2}$$

Expanding the expression for BC:

$$BC^2 = (c - d)^2 + e^2$$ $$= c^2 - 2cd + d^2 + e^2$$

Setting up the Law of Cosines:

$$BC^2 = AB^2 + AC^2 - 2 \cdot AB \cdot AC \cdot \cos(\angle BAC)$$ $$c^2 - 2cd + d^2 + e^2 = c^2 + d^2 + e^2 - 2c\sqrt{d^2 + e^2}\cos(\angle BAC)$$

Simplifying, we find:

$$-2cd = -2c\sqrt{d^2 + e^2}\cos(\angle BAC)$$ $$\cos(\angle BAC) = \frac{d}{\sqrt{d^2 + e^2}}$$

This confirms the validity of the Law of Cosines through coordinate geometry.

Complex Problem-Solving with the Law of Cosines

Consider a triangle with sides of lengths 13 units, 14 units, and 15 units. Determine all three angles using the Law of Cosines.

Solution:

• Let sides a = 13, b = 14, and c = 15.
• Find angle C using:
$$c^2 = a^2 + b^2 - 2ab\cos(C)$$ $$225 = 169 + 196 - 2 \times 13 \times 14 \cos(C)$$ $$225 = 365 - 364\cos(C)$$ $$364\cos(C) = 365 - 225$$ $$364\cos(C) = 140$$ $$\cos(C) = \frac{140}{364} = \frac{35}{91} \approx 0.3846$$ $$C \approx \cos^{-1}(0.3846) \approx 67.5^\circ$$
• Find angle B using:
$$b^2 = a^2 + c^2 - 2ac\cos(B)$$ $$196 = 169 + 225 - 2 \times 13 \times 15 \cos(B)$$ $$196 = 394 - 390\cos(B)$$ $$390\cos(B) = 394 - 196$$ $$390\cos(B) = 198$$ $$\cos(B) = \frac{198}{390} = \frac{33}{65} \approx 0.5077$$ $$B \approx \cos^{-1}(0.5077) \approx 59.5^\circ$$
• Find angle A using the fact that the sum of angles in a triangle is 180 degrees:
$$A = 180^\circ - B - C$$ $$A = 180^\circ - 59.5^\circ - 67.5^\circ$$ $$A = 53^\circ$$

Thus, the angles are approximately:

• A ≈ 53°
• B ≈ 59.5°
• C ≈ 67.5°

Interdisciplinary Connections

The Law of Cosines is not confined to pure mathematics; it has significant applications across various disciplines:

• Engineering: Used in structural analysis to determine component forces and stresses in truss structures.
• Computer Science: Essential in computer graphics for calculating angles and distances between objects in 3D space.
• Physics: Applied in resolving vector quantities such as force, velocity, and acceleration in multiple dimensions.
• Geography: Utilized in geographic information systems (GIS) to compute distances between geographic coordinates.

Proof Using the Cosine Rule in Vector Spaces

In vector spaces, the Law of Cosines can be interpreted through the dot product. Given vectors u and v, the dot product is defined as:

$$\mathbf{u} \cdot \mathbf{v} = ||\mathbf{u}|| \cdot ||\mathbf{v}|| \cdot \cos(\theta)$$

Where θ is the angle between the vectors. To find the magnitude of the sum of the vectors:

$$||\mathbf{u} + \mathbf{v}||^2 = (\mathbf{u} + \mathbf{v}) \cdot (\mathbf{u} + \mathbf{v})$$ $$= \mathbf{u} \cdot \mathbf{u} + 2\mathbf{u} \cdot \mathbf{v} + \mathbf{v} \cdot \mathbf{v}$$ $$= ||\mathbf{u}||^2 + 2||\mathbf{u}|| \cdot ||\mathbf{v}|| \cdot \cos(\theta) + ||\mathbf{v}||^2$$

This aligns with the Law of Cosines formula, demonstrating its validity within vector spaces.

Complex Geometric Constructions

The Law of Cosines aids in constructing geometric shapes and solving complex geometric problems. For example, determining the height of an oblique triangle or calculating the area of a triangle when all three sides are known can be efficiently achieved using this law.

Given a triangle with sides a, b, and c, the area can be found using Heron's formula. However, incorporating the Law of Cosines provides an alternative approach:

$$Area = \frac{1}{2}ab\sin(C)$$

By first finding angle C using the Law of Cosines, one can substitute the value into the area formula to compute the triangle's area.

Comparison Table

Aspect	Law of Sines	Law of Cosines
Primary Use	Solving triangles when two angles and one side (AAS) or two sides and a non-included angle (SSA) are known.	Solving triangles when two sides and the included angle (SAS) or three sides (SSS) are known.
Formula	$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$	$c^2 = a^2 + b^2 - 2ab\cos(C)$
Applicability	All types of triangles, especially useful for acute and obtuse triangles.	All types of triangles, including right-angled triangles as a special case.
Advantages	Simpler formulas for certain scenarios; easier to apply when angles are involved.	Versatile for various scenarios, including cases where the triangle is not easily solvable by the Law of Sines.
Limitations	May lead to ambiguous cases (SSA) where two different triangles satisfy the given conditions.	Requires knowledge of included angles; more complex calculations for unknown angles.

Summary and Key Takeaways