Distance Formula & Fundamental Laws
Distance Formula
The distance between two points A(x₁, y₁) and B(x₂, y₂) is given by:
Fundamental Law of Trigonometry
For any two angles α and β (any real numbers):
This is the foundation from which all other trigonometric identities can be derived.
The distance formula is actually derived from the Pythagorean theorem! The fundamental law cos(α-β) is the cornerstone of trigonometric identities – memorize it well.
Problem-Solving Power
Advanced trigonometry formulas enable you to solve complex problems in physics, engineering, and advanced mathematics that basic trigonometry cannot handle.
Wave Analysis
Transformation formulas are essential for analyzing waveforms, signals, and oscillations in physics, engineering, and computer graphics.
Calculus Foundation
These identities are crucial for integration and differentiation of trigonometric functions in calculus, making them essential for advanced mathematics.
Sum & Difference Formulas
cos(α ± β) = cos α cos β ∓ sin α sin β
tan(α ± β) = (tan α ± tan β)/(1 ∓ tan α tan β)
For sine: “Same operation” (sin cos ± cos sin). For cosine: “Different operation” (cos cos ∓ sin sin). The sign in the formula matches the sign in the argument for sine, but is opposite for cosine.
Double Angle Formulas
cos 2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ
tan 2θ = 2tanθ/(1 – tan²θ)
sin 2θ = 2 sin θ cos θ (just double it!)
cos 2θ has three forms – choose based on what’s given:
1. cos²θ – sin²θ (basic)
2. 2cos²θ – 1 (if you know cos θ)
3. 1 – 2sin²θ (if you know sin θ)
Half Angle Formulas
cos(θ/2) = ±√[(1 + cos θ)/2]
tan(θ/2) = ±√[(1 – cos θ)/(1 + cos θ)] = sin θ/(1 + cos θ) = (1 – cos θ)/sin θ
The ± sign depends on the quadrant of θ/2. Remember: For half-angle formulas, you need to know the quadrant of the half-angle, not the original angle!
Triple Angle Formulas
cos 3θ = 4 cos³θ – 3 cos θ
tan 3θ = (3 tan θ – tan³θ)/(1 – 3 tan²θ)
Notice the pattern: For sin 3θ, coefficients are 3 and 4. For cos 3θ, coefficients are 4 and 3. The powers are cubes. This symmetry helps in memorization.
Transformation Formulas
Product to Sum/Difference:
2 cos A sin B = sin(A+B) – sin(A-B)
2 cos A cos B = cos(A+B) + cos(A-B)
2 sin A sin B = cos(A-B) – cos(A+B)
Sum/Difference to Product:
sin C – sin D = 2 cos[(C+D)/2] sin[(C-D)/2]
cos C + cos D = 2 cos[(C+D)/2] cos[(C-D)/2]
cos C – cos D = -2 sin[(C+D)/2] sin[(C-D)/2]
For product to sum: “2 sin cos gives sin+sin”, “2 cos sin gives sin-sin”, “2 cos cos gives cos+cos”, “2 sin sin gives cos-cos” (with sign adjustment).
Allied Angles
Angles associated with basic angle θ to a right angle or its multiples:
- sin(π/2 ± θ) = cos θ
- cos(π/2 ± θ) = ∓ sin θ
- sin(π ± θ) = ∓ sin θ
- cos(π ± θ) = -cos θ
- sin(3π/2 ± θ) = -cos θ
- cos(3π/2 ± θ) = ± sin θ
- sin(2π ± θ) = ± sin θ
- cos(2π ± θ) = cos θ
1. Odd multiples of π/2: Functions change to co-functions
2. Even multiples of π/2: Functions remain the same
3. Determine sign based on quadrant of resulting angle
Maximum & Minimum Values
For expression: a sin θ + b cos θ
Minimum value = -√(a² + b²)
This occurs when sin(θ + φ) = 1 (for max) or -1 (for min), where tan φ = b/a.
Think of a sin θ + b cos θ as the projection of a vector of length √(a²+b²) rotating in a circle. The maximum projection is the full length, minimum is negative of that length.
Pythagorean Identities
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ
These are derived from the fundamental identity sin²θ + cos²θ = 1 by dividing by cos²θ or sin²θ.
From sin²θ + cos²θ = 1:
1. Divide by cos²θ: tan²θ + 1 = sec²θ
2. Divide by sin²θ: 1 + cot²θ = csc²θ
Test Your Advanced Trigonometry Knowledge
50 Multiple Choice Questions | Time: Unlimited | Score: 2 points per question
Instructions: Click on the correct answer. After selection, you’ll see immediate feedback. Click “Next Question” to continue.
Understand Derivations
Don’t just memorize formulas – understand how they are derived from fundamental identities. This helps in recalling them during exams and applying them to novel problems.
Group Related Formulas
Organize formulas in logical groups: sum/difference, double/half angles, product/sum transformations, etc. Create mind maps showing relationships between different formulas.
Practice Derivation
Regularly practice deriving important formulas from basic principles. For example, derive double angle formulas from sum formulas, or half angle formulas from double angle formulas.
Apply Transformation Formulas
Practice converting products to sums and vice versa. These skills are crucial for integration in calculus and solving trigonometric equations.
Master Allied Angles
Create a quadrant-based chart for allied angles. Practice determining signs for angles in different quadrants without memorizing each case individually.
Practice Maximum/Minimum Problems
Work on problems finding extreme values of expressions like a sin θ + b cos θ. Visualize these as rotating vectors to build intuition.
Solve Complex Equations
Practice solving trigonometric equations that require multiple identities. Break them down step by step, identifying which identities to apply at each stage.
Time Yourself
When practicing, time yourself to build speed. Advanced trigonometry problems in exams often require quick recall and application of multiple formulas.
Verify with Multiple Methods
When possible, solve problems using different approaches or formulas to verify your answer. This deepens understanding and builds confidence.
Create Formula Sheets
Make your own comprehensive formula sheets organized by topic. The act of creating them helps with memorization and understanding relationships.