Trigonometry Basics – Core Formulas and Identities

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Pythagorean Identities - The Foundation

\[ \sin^2 \theta + \cos^2 \theta = 1 \]

\[ 1 + \tan^2 \theta = \sec^2 \theta \]

\[ 1 + \cot^2 \theta = \csc^2 \theta \]

\[ \text{These are the fundamental relationships derived from the unit circle} \]

🎯 What does this mean?

Fundamental trigonometric identities are the essential mathematical relationships that exist between trigonometric functions for all valid angle values. These identities represent the unchanging patterns and connections within trigonometry, derived from the geometric properties of the unit circle and right triangles. They serve as the mathematical "toolbox" that enables simplification of complex expressions, solving of trigonometric equations, and transformation between different trigonometric forms. Mastering these basic formulas is essential for all advanced trigonometric work, calculus applications, and real-world problem-solving.

📊 Fundamental Trigonometric Identities from Reference

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Core Pythagorean Identity

\[ \sin^2 \alpha + \cos^2 \alpha = 1 \]

\[ \text{Explanation: This is the fundamental Pythagorean identity. It arises from the Pythagorean theorem applied to a unit circle, where the radius (hypotenuse) is 1.} \]

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Reciprocal Identity - Tangent and Cotangent

\[ \tan \alpha \cdot \cot \alpha = 1 \]

\[ \text{Explanation: Tangent and cotangent are reciprocals of each other. So, multiplying them always results in 1: } \tan \alpha = \frac{\sin \alpha}{\cos \alpha}, \cot \alpha = \frac{\cos \alpha}{\sin \alpha} \]

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Quotient Identity - Tangent

\[ \tan \alpha = \frac{\sin \alpha}{\cos \alpha} = \frac{1}{\cot \alpha} \]

\[ \text{Explanation: This is a quotient identity. Tangent is the ratio of sine to cosine, and it is the reciprocal of cotangent.} \]

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Quotient Identity - Cotangent

\[ \cot \alpha = \frac{\cos \alpha}{\sin \alpha} = \frac{1}{\tan \alpha} \]

\[ \text{Explanation: Cotangent is the reciprocal of tangent. It's defined as the ratio of cosine to sine.} \]

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Pythagorean Identity - Tangent and Secant

\[ 1 + \tan^2 \alpha = \frac{1}{\cos^2 \alpha} = \sec^2 \alpha \]

\[ \text{Explanation: This identity comes from dividing the Pythagorean identity by } \cos^2 \alpha. \text{ It links tangent and secant functions.} \]

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Pythagorean Identity - Cotangent and Cosecant

\[ 1 + \cot^2 \alpha = \frac{1}{\sin^2 \alpha} = \csc^2 \alpha \]

\[ \text{Explanation: This identity is derived by dividing the Pythagorean identity by } \sin^2 \alpha, \text{ relating cotangent and cosecant.} \]

Basic Trigonometric Formulas
• These identities define key relationships between trigonometric functions
• Essential in simplifying, solving, and transforming trigonometric equations
• Derived from unit circle and Pythagorean theorem principles
• Form the foundation for all advanced trigonometric work

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Reciprocal Identities

Fundamental reciprocal relationships:

\[ \sin \theta = \frac{1}{\csc \theta} \quad \text{and} \quad \csc \theta = \frac{1}{\sin \theta} \]

\[ \cos \theta = \frac{1}{\sec \theta} \quad \text{and} \quad \sec \theta = \frac{1}{\cos \theta} \]

\[ \tan \theta = \frac{1}{\cot \theta} \quad \text{and} \quad \cot \theta = \frac{1}{\tan \theta} \]

\[ \text{Note: Functions undefined when denominator equals zero} \]

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Quotient Identities

Relationships expressing tangent and cotangent as ratios:

\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]

\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \]

\[ \text{Restrictions: } \cos \theta \neq 0 \text{ for tangent, } \sin \theta \neq 0 \text{ for cotangent} \]

\[ \text{These connect the six trigonometric functions} \]

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Co-function Identities

Relationships between complementary angles:

\[ \sin \theta = \cos(90° - \theta) = \cos\left(\frac{\pi}{2} - \theta\right) \]

\[ \cos \theta = \sin(90° - \theta) = \sin\left(\frac{\pi}{2} - \theta\right) \]

\[ \tan \theta = \cot(90° - \theta) = \cot\left(\frac{\pi}{2} - \theta\right) \]

\[ \sec \theta = \csc(90° - \theta), \quad \csc \theta = \sec(90° - \theta) \]

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Even-Odd Function Properties

Symmetry properties of trigonometric functions:

\[ \sin(-\theta) = -\sin \theta \quad \text{(sine is odd)} \]

\[ \cos(-\theta) = \cos \theta \quad \text{(cosine is even)} \]

\[ \tan(-\theta) = -\tan \theta \quad \text{(tangent is odd)} \]

\[ \text{Similar patterns: } \csc(-\theta) = -\csc \theta, \sec(-\theta) = \sec \theta, \cot(-\theta) = -\cot \theta \]

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Periodicity Identities

Repeating patterns of trigonometric functions:

\[ \sin(\theta + 2\pi) = \sin \theta \quad \text{(period of } 2\pi \text{)} \]

\[ \cos(\theta + 2\pi) = \cos \theta \quad \text{(period of } 2\pi \text{)} \]

\[ \tan(\theta + \pi) = \tan \theta \quad \text{(period of } \pi \text{)} \]

\[ \text{Similar: } \cot(\theta + \pi) = \cot \theta, \sec(\theta + 2\pi) = \sec \theta, \csc(\theta + 2\pi) = \csc \theta \]

🎯

Fundamental Angle Values

Essential exact values for common angles:

\[ \sin 0° = 0, \sin 30° = \frac{1}{2}, \sin 45° = \frac{\sqrt{2}}{2}, \sin 60° = \frac{\sqrt{3}}{2}, \sin 90° = 1 \]

\[ \cos 0° = 1, \cos 30° = \frac{\sqrt{3}}{2}, \cos 45° = \frac{\sqrt{2}}{2}, \cos 60° = \frac{1}{2}, \cos 90° = 0 \]

\[ \tan 0° = 0, \tan 30° = \frac{\sqrt{3}}{3}, \tan 45° = 1, \tan 60° = \sqrt{3}, \tan 90° = \text{undefined} \]

\[ \text{Radian equivalents: } 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2} \]

⚠️

Common Mistakes and Memory Aids

Critical points for accurate application:

\[ \sin^2 \theta + \cos^2 \theta = 1 \text{ (NOT } \sin \theta + \cos \theta = 1 \text{)} \]

\[ \text{Reciprocals: } \sin \theta \cdot \csc \theta = 1 \text{ (when defined)} \]

\[ \text{Domain restrictions: Watch for zeros in denominators} \]

\[ \text{Memorize unit circle values for quick reference} \]

\[ \text{From screenshot: Verify derivations using Pythagorean theorem} \]

🎯 Mathematical Interpretation

Fundamental trigonometric identities represent the unchanging mathematical relationships that govern the behavior of trigonometric functions. These identities emerge from the geometric properties of the unit circle and the algebraic structure of trigonometric ratios. They form the logical foundation that connects all trigonometric functions and enable transformation between different mathematical expressions. Understanding these identities as universal truths rather than mere formulas allows students to see the elegant patterns within trigonometry and develop the algebraic fluency needed for advanced mathematics, physics, and engineering applications. The screenshot demonstrates how these identities derive from the Pythagorean theorem and unit circle properties.

\[ \sin^2 \alpha + \cos^2 \alpha = 1 \]

Fundamental Pythagorean Identity - derived from unit circle and Pythagorean theorem

\[ \tan \alpha \cdot \cot \alpha = 1 \]

Reciprocal relationship - tangent and cotangent multiply to give 1

\[ \tan \alpha = \frac{\sin \alpha}{\cos \alpha} \]

Quotient identity - tangent as ratio of sine to cosine

\[ \cot \alpha = \frac{\cos \alpha}{\sin \alpha} \]

Cotangent quotient - reciprocal of tangent as cosine over sine

\[ 1 + \tan^2 \alpha = \sec^2 \alpha \]

Extended Pythagorean - linking tangent and secant functions

\[ 1 + \cot^2 \alpha = \csc^2 \alpha \]

Cotangent-cosecant relationship - derived from Pythagorean identity

\[ \text{Reciprocal Pairs} \]

Sin-Csc, Cos-Sec, Tan-Cot - functions that multiply to give 1

\[ \text{Co-functions} \]

Complementary angle relationships - sine becomes cosine, etc.

\[ \text{Even/Odd} \]

Symmetry properties - cosine is even, sine and tangent are odd

\[ \text{Periodicity} \]

Repeating patterns - 2π for sine/cosine, π for tangent/cotangent

\[ \text{Unit Circle} \]

Geometric foundation - all identities derive from circle properties

\[ \text{Domain Restrictions} \]

Undefined points - where denominators equal zero in ratios

🎯 Essential Insight: Fundamental identities are the "universal constants" of trigonometry - they're like mathematical DNA that never changes, no matter what angle you're working with! 🧬

🚀 Real-World Applications

⚡ Electrical Engineering & Power Systems

AC Circuit Analysis & Power Calculations

Engineers use fundamental identities for analyzing alternating current, calculating power factors, designing transformers, and optimizing electrical grid performance

🌊 Physics & Wave Mechanics

Oscillation Analysis & Wave Functions

Physicists apply identities for analyzing simple harmonic motion, electromagnetic waves, quantum mechanics, and signal processing in various physical systems

🏗️ Structural Engineering & Architecture

Force Analysis & Structural Design

Engineers use identities for calculating forces in trusses, designing earthquake-resistant buildings, analyzing structural vibrations, and optimizing load distributions

📡 Communications & Signal Processing

Digital Signal Analysis & Modulation

Engineers apply identities for Fourier analysis, digital filtering, radio wave propagation, satellite communications, and audio/video compression algorithms

The Magic: Electrical: AC circuit analysis and power systems, Physics: Wave mechanics and oscillations, Engineering: Structural force analysis, Communications: Signal processing and modulation

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Master the "Identity Arsenal" Method!

Before tackling any trigonometric problem, build your fundamental identity toolkit:

Key Insight: Fundamental trigonometric identities are like having a universal translator for the language of trigonometry! Think of yourself as a mathematical linguist with an identity arsenal that can transform any trigonometric expression into the form you need. Master these basic patterns and you'll speak trigonometry fluently!

💡 Why this matters:

🔋 Real-World Power:

Electrical: AC circuit analysis and power calculations
Physics: Wave mechanics and oscillation analysis
Engineering: Structural design and force analysis
Communications: Signal processing and digital modulation

🧠 Mathematical Insight:

Universal relationships: Identities work for ANY angle value
Expression simplification: Transform complex forms into manageable expressions
Equation solving: Convert trigonometric equations to solvable forms
Geometric foundation: All derive from unit circle and Pythagorean theorem

🚀 Study Strategy:

1 Master the Core Arsenal 📚

Pythagorean: sin²θ + cos²θ = 1 (the fundamental identity)
Reciprocal: sin·csc = 1, cos·sec = 1, tan·cot = 1
Quotient: tan = sin/cos, cot = cos/sin
Memory technique: Practice until these are automatic reflexes

2 Understand the Derivations 🔍

Pythagorean identity from unit circle (radius = 1)
Extended forms: 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ
Reciprocal relationships from function definitions
Connect algebraic forms to geometric meanings

3 Apply Strategic Substitution 📊

Identify which identity best simplifies the expression
Use Pythagorean identities to eliminate squared terms
Apply reciprocal identities to convert between functions
Use quotient identities to work with single functions

4 Master Advanced Applications 🔗

Combine identities for complex simplifications
Use identities to solve trigonometric equations
Apply in calculus for integration and differentiation
Connect to real-world problems in physics and engineering

When you master the "identity arsenal" approach and understand fundamental identities as universal mathematical patterns derived from geometric principles, you'll have the power to simplify any trigonometric expression, solve complex equations, tackle calculus problems, and apply trigonometry to real-world applications in electrical engineering, wave physics, structural analysis, and signal processing!

Memory Trick: "Universal Patterns" - PYTHAGOREAN: sin² + cos² = 1, RECIPROCAL: Multiply to get 1, QUOTIENT: Tangent is sine over cosine

🔑 Key Properties of Fundamental Identities

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Universal Truth

Identities hold true for all valid angle values without exception

They represent unchanging mathematical relationships independent of specific calculations

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Geometric Foundation

All identities derive from unit circle and Pythagorean theorem properties

Connect algebraic manipulation with geometric understanding

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Bidirectional Use

Can be applied in both directions for expression transformation

Enable both simplification and strategic complication when needed

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Interconnected System

All six trigonometric functions are related through these identities

Mastering one set enables understanding of all trigonometric relationships

Universal Insight: Fundamental identities are mathematical constants derived from geometric principles - they're the DNA of trigonometry!

General Approach: Memorize core identities, understand derivations, apply strategic substitution

Essential Memory: Pythagorean (sin² + cos² = 1), Reciprocal (multiply to get 1), Quotient (tan = sin/cos)

Key Tools: Identity substitution, pattern recognition, algebraic manipulation, and geometric understanding

Applications: Circuit analysis, wave mechanics, structural engineering, and signal processing

Basic Trigonometric Formulas – Identities and Relations