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Cosine Law (Law of Cosines)

Master the fundamental cosine law for triangle solving: understand the relationship between sides and included angles, learn calculation methods for unknown sides and angles, and apply to complex triangle problems and real-world applications.

Fundamental Definitions and Formulas
Essential cosine law concepts and mathematical expressions
Standard Cosine Law Forms
Fundamental relationships between sides and angles in any triangle
Primary Formula:
a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2 - 2bc\cos A
b2=a2+c2−2accos⁡Bc2=a2+b2−2abcos⁡Cb^2 = a^2 + c^2 - 2ac\cos B \quad c^2 = a^2 + b^2 - 2ab\cos C
Key Properties:
  • Relates three sides and one angle in a triangle
  • Generalizes the Pythagorean theorem to any triangle
  • When angle is 90°, reduces to Pythagorean theorem
  • Essential for solving SAS and SSS triangle cases
Angle-Finding Forms
Rearranged cosine law to find angles from known sides
Primary Formula:
cos⁡A=b2+c2−a22bc\cos A = \frac{b^2 + c^2 - a^2}{2bc}
cos⁡B=a2+c2−b22accos⁡C=a2+b2−c22ab\cos B = \frac{a^2 + c^2 - b^2}{2ac} \quad \cos C = \frac{a^2 + b^2 - c^2}{2ab}
Key Properties:
  • Calculates angles when all three sides are known
  • Ensures angles are always between 0° and 180°
  • Cosine value determines triangle type (acute/obtuse)
  • Foundation for complete triangle analysis
Geometric Interpretation
Understanding cosine law through geometric relationships
Primary Formula:
a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2 - 2bc\cos A
The term −2bccos⁡A represents the ’correction’ to Pythagorean theorem\text{The term } -2bc\cos A \text{ represents the 'correction' to Pythagorean theorem}
Key Properties:
  • When A = 90°: cos A = 0, giving Pythagorean theorem
  • When A < 90°: cos A > 0, so a² < b² + c² (acute triangle)
  • When A > 90°: cos A < 0, so a² > b² + c² (obtuse triangle)
  • Relates triangle geometry to algebraic expressions
Triangle Solving Applications
Common scenarios for applying cosine law
Case 1: Side-Angle-Side (SAS)
Given two sides and the included angle, find the third side
Approach:
Direct application of standard cosine law
Solution Steps:
  1. 1Identify the two known sides and included angle
  2. 2Apply cosine law: a² = b² + c² - 2bc cos A
  3. 3Calculate the value step by step
  4. 4Take square root to find the third side
Example: Given: b = 8, c = 6, A = 60°. Find: a
Case 2: Side-Side-Side (SSS)
Given all three sides, find any angle
Approach:
Use angle-finding form of cosine law
Solution Steps:
  1. 1Choose which angle to find first
  2. 2Apply angle form: cos A = (b² + c² - a²)/(2bc)
  3. 3Calculate cosine value
  4. 4Use inverse cosine to find the angle
Example: Given: a = 7, b = 8, c = 9. Find: angle A
Case 3: Completing Triangle Solutions
Use cosine law with other triangle properties
Approach:
Combine with angle sum theorem and sine law
Solution Steps:
  1. 1Use cosine law to find one unknown element
  2. 2Apply angle sum: A + B + C = 180°
  3. 3Use sine law for remaining elements if needed
  4. 4Verify all results are consistent
Example: Solve triangle completely using mixed information
Triangle Classification Using Cosine Law
Determining triangle types from cosine values
Acute Triangle
Mathematical Condition:
cos⁡A>0,cos⁡B>0,cos⁡C>0\cos A > 0, \cos B > 0, \cos C > 0
Cosine Signature:
All cosine values are positive
Side Relationship:
a² < b² + c² for all sides
All angles are less than 90°
Properties:
  • All angles acute (< 90°)
  • Each side squared is less than sum of other two sides squared
  • Circumcenter lies inside the triangle
  • All heights lie inside the triangle
Right Triangle
Mathematical Condition:
cos⁡A=0 (when A = 90°)\cos A = 0 \text{ (when A = 90°)}
Cosine Signature:
One cosine equals zero, others positive
Side Relationship:
a² = b² + c² (Pythagorean theorem)
Exactly one angle equals 90°
Properties:
  • One angle exactly 90°, others acute
  • Pythagorean theorem applies
  • Circumcenter lies on hypotenuse
  • Special trigonometric relationships
Obtuse Triangle
Mathematical Condition:
cos⁡A<0 (when A > 90°)\cos A < 0 \text{ (when A > 90°)}
Cosine Signature:
One cosine negative, others positive
Side Relationship:
a² > b² + c² for the obtuse angle's opposite side
Exactly one angle is greater than 90°
Properties:
  • One angle obtuse (> 90°), others acute
  • Longest side opposite the obtuse angle
  • Circumcenter lies outside the triangle
  • Special geometric properties
Triangle Area and Cosine Law
Connecting area calculations with cosine law
SAS Area Formula
Formula:
S=12bcsin⁡A=12acsin⁡B=12absin⁡CS = \frac{1}{2}bc\sin A = \frac{1}{2}ac\sin B = \frac{1}{2}ab\sin C
Area using two sides and included angle
Connection to Cosine Law: Complements cosine law for complete triangle analysis
Heron's Formula
Formula:
S=s(s−a)(s−b)(s−c) where s=a+b+c2S = \sqrt{s(s-a)(s-b)(s-c)} \text{ where } s = \frac{a+b+c}{2}
Area using all three sides only
Connection to Cosine Law: Alternative when only sides are known (SSS case)
Combined Approach
Formula:
Use cosine law to find angles, then apply SAS area formula\text{Use cosine law to find angles, then apply SAS area formula}
Two-step process for SSS triangles
Connection to Cosine Law: Most systematic approach for complete solutions
Problem-Solving Strategies
Systematic approaches to cosine law problems
Finding Unknown Sides
Apply standard cosine law formula
Methodology:
When given SAS configuration
Step-by-step Process:
  1. 1Verify you have two sides and included angle
  2. 2Substitute into a² = b² + c² - 2bc cos A
  3. 3Calculate 2bc cos A carefully
  4. 4Compute a² then take positive square root
Finding Unknown Angles
Use rearranged cosine law for angles
Methodology:
When given SSS configuration or SAS seeking other angles
Step-by-step Process:
  1. 1Ensure all three sides are known
  2. 2Apply cos A = (b² + c² - a²)/(2bc)
  3. 3Calculate the fraction carefully
  4. 4Use inverse cosine to find angle
Complete Triangle Analysis
Systematic application of multiple laws
Methodology:
Comprehensive solution strategy
Step-by-step Process:
  1. 1Identify given information and required unknowns
  2. 2Choose most appropriate law (cosine vs sine)
  3. 3Solve step-by-step, checking consistency
  4. 4Verify results using triangle properties
Real-World Applications
How cosine law is used in various fields
Engineering and Construction
  • Structural framework analysis and design
  • Bridge construction and load calculations
  • Roof truss design and angle determination
  • Mechanical linkage analysis
Navigation and Surveying
  • GPS triangulation and position fixing
  • Maritime navigation and course plotting
  • Land surveying and property boundaries
  • Aircraft navigation and flight planning
Physics and Astronomy
  • Vector addition and resultant calculations
  • Celestial mechanics and orbital calculations
  • Wave interference and phase relationships
  • Crystallography and molecular geometry
Computer Graphics and Gaming
  • 3D graphics and polygon rendering
  • Game physics and collision detection
  • Animation and interpolation calculations
  • Virtual reality and spatial relationships
Advanced Applications and Extensions
Beyond basic triangle solving
Extended Law of Cosines
Generalization to spherical triangles and higher dimensions
Extended Formula:
For spherical triangles: cos⁡a=cos⁡bcos⁡c+sin⁡bsin⁡ccos⁡A\text{For spherical triangles: } \cos a = \cos b \cos c + \sin b \sin c \cos A
Applications:
  • Spherical geometry
  • Navigation on Earth's surface
  • Astronomy calculations
Vector Form of Cosine Law
Expressing cosine law using vector dot products
Extended Formula:
c⃗⋅c⃗=(a⃗−b⃗)⋅(a⃗−b⃗)=∣a⃗∣2+∣b⃗∣2−2a⃗⋅b⃗\vec{c} \cdot \vec{c} = (\vec{a} - \vec{b}) \cdot (\vec{a} - \vec{b}) = |\vec{a}|^2 + |\vec{b}|^2 - 2\vec{a} \cdot \vec{b}
Applications:
  • Vector analysis
  • Physics applications
  • Computer graphics
Cosine Law in Complex Analysis
Application to complex numbers and plane geometry
Extended Formula:
∣z1−z2∣2=∣z1∣2+∣z2∣2−2Re(z1z2‾)|z_1 - z_2|^2 = |z_1|^2 + |z_2|^2 - 2\text{Re}(z_1\overline{z_2})
Applications:
  • Complex plane geometry
  • Signal processing
  • Electrical engineering
Important Points to Remember
Common misconceptions and key insights
✓ Key Insights
Pythagorean Extension: Cosine law generalizes the Pythagorean theorem to all triangles, not just right triangles.
Triangle Type Detection: The sign of cosine values immediately reveals whether a triangle is acute, right, or obtuse.
⚠ Common Mistakes
Side-Angle Correspondence: Ensure the angle is between the two given sides in SAS problems - otherwise use sine law.
Calculator Mode: Check that your calculator is in the correct mode (degrees vs radians) before computing inverse cosine.