1. Introduction to Trigonometry
Trigonometry is the branch of mathematics that studies relationships between angles and sides of triangles, particularly right triangles.
Key Concepts
- Angle: Rotation from one ray to another
- Vertex: Point where two rays meet
- Standard Position: Angle with vertex at origin, initial side on positive x-axis
- Quadrants: Four regions of coordinate plane
- Reference Angle: Acute angle between terminal side and x-axis
2. Angle Measurement
Degree Measure
- Full rotation: 360°
- Straight angle: 180°
- Right angle: 90°
- Acute angle: 0° < θ < 90°
- Obtuse angle: 90° < θ < 180°
Radian Measure
- Definition: Arc length equal to radius
- Full rotation: 2π radians
- Conversion: π radians = 180°
- Formula: radians = (π/180°) × degrees
- Formula: degrees = (180°/π) × radians
Common Angle Conversions
| Degrees | Radians |
|---|
| 0° | 0 |
| 30° | π/6 |
| 45° | π/4 |
| 60° | π/3 |
| 90° | π/2 |
| 180° | π |
| 270° | 3π/2 |
| 360° | 2π |
3. Trigonometric Ratios in Right Triangles
Basic Ratios (SOH-CAH-TOA)
For angle θ in a right triangle:
- sin θ = Opposite/Hypotenuse (SOH)
- cos θ = Adjacent/Hypotenuse (CAH)
- tan θ = Opposite/Adjacent (TOA)
Reciprocal Ratios
- csc θ = 1/sin θ = Hypotenuse/Opposite
- sec θ = 1/cos θ = Hypotenuse/Adjacent
- cot θ = 1/tan θ = Adjacent/Opposite
Pythagorean Relationship
- sin²θ + cos²θ = 1 (Fundamental Identity)
4. Special Right Triangles
45°-45°-90° Triangle
- Sides ratio: 1 : 1 : √2
- sin 45° = cos 45° = 1/√2 = √2/2
- tan 45° = 1
30°-60°-90° Triangle
- Sides ratio: 1 : √3 : 2
- sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3
- sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3
Standard Values Table
| Angle | sin | cos | tan |
|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
5. Trigonometric Functions for All Angles
Unit Circle
- Radius: 1
- Point on circle: (cos θ, sin θ)
- Distance from origin: Always 1
Signs in Quadrants
| Quadrant | sin | cos | tan |
|---|
| I (0° to 90°) | + | + | + |
| II (90° to 180°) | + | – | – |
| III (180° to 270°) | – | – | + |
| IV (270° to 360°) | – | + | – |
Memory Aid: “All Students Take Calculus” (All positive, Sin positive, Tan positive, Cos positive)
Reference Angles
- Quadrant II: 180° – θ
- Quadrant III: θ – 180°
- Quadrant IV: 360° – θ
6. Fundamental Trigonometric Identities
Pythagorean Identities
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = csc²θ
Reciprocal Identities
- sin θ = 1/csc θ
- cos θ = 1/sec θ
- tan θ = 1/cot θ
Quotient Identities
- tan θ = sin θ/cos θ
- cot θ = cos θ/sin θ
Co-function Identities
- sin θ = cos(90° – θ)
- cos θ = sin(90° – θ)
- tan θ = cot(90° – θ)
Even-Odd Identities
- sin(-θ) = -sin θ (odd function)
- cos(-θ) = cos θ (even function)
- tan(-θ) = -tan θ (odd function)
7. Sum and Difference Formulas
Sine Formulas
- sin(A + B) = sin A cos B + cos A sin B
- sin(A – B) = sin A cos B – cos A sin B
Cosine Formulas
- cos(A + B) = cos A cos B – sin A sin B
- cos(A – B) = cos A cos B + sin A sin B
Tangent Formulas
- tan(A + B) = (tan A + tan B)/(1 – tan A tan B)
- tan(A – B) = (tan A – tan B)/(1 + tan A tan B)
8. Double Angle Formulas
Sine Double Angle
Cosine Double Angle
- cos 2θ = cos²θ – sin²θ
- cos 2θ = 2cos²θ – 1
- cos 2θ = 1 – 2sin²θ
Tangent Double Angle
- tan 2θ = (2 tan θ)/(1 – tan²θ)
9. Half Angle Formulas
Sine Half Angle
- sin²(θ/2) = (1 – cos θ)/2
- sin(θ/2) = ±√[(1 – cos θ)/2]
Cosine Half Angle
- cos²(θ/2) = (1 + cos θ)/2
- cos(θ/2) = ±√[(1 + cos θ)/2]
Tangent Half Angle
- tan(θ/2) = (1 – cos θ)/sin θ = sin θ/(1 + cos θ)
10. Product-to-Sum Formulas
- sin A cos B = ½[sin(A + B) + sin(A – B)]
- cos A sin B = ½[sin(A + B) – sin(A – B)]
- cos A cos B = ½[cos(A + B) + cos(A – B)]
- sin A sin B = ½[cos(A – B) – cos(A + B)]
11. Sum-to-Product Formulas
- sin A + sin B = 2 sin[(A + B)/2] cos[(A – B)/2]
- sin A – sin B = 2 cos[(A + B)/2] sin[(A – B)/2]
- cos A + cos B = 2 cos[(A + B)/2] cos[(A – B)/2]
- cos A – cos B = -2 sin[(A + B)/2] sin[(A – B)/2]
12. Graphs of Trigonometric Functions
Sine Function: y = sin x
- Domain: All real numbers
- Range: [-1, 1]
- Period: 2π
- Amplitude: 1
- Key points: (0,0), (π/2,1), (π,0), (3π/2,-1), (2π,0)
Cosine Function: y = cos x
- Domain: All real numbers
- Range: [-1, 1]
- Period: 2π
- Amplitude: 1
- Key points: (0,1), (π/2,0), (π,-1), (3π/2,0), (2π,1)
Tangent Function: y = tan x
- Domain: All real numbers except odd multiples of π/2
- Range: All real numbers
- Period: π
- Vertical asymptotes: x = π/2 + nπ
Transformations
For y = A sin(Bx + C) + D:
- A: Amplitude (vertical stretch)
- B: Affects period (Period = 2π/|B|)
- C: Phase shift (horizontal shift = -C/B)
- D: Vertical shift
13. Solving Trigonometric Equations
Basic Strategy
- Isolate the trigonometric function
- Find reference angle
- Determine all angles in given interval
- Check solutions
Types of Solutions
- Principal values: Values in specific intervals
- General solutions: All possible solutions
- Specific interval solutions: Solutions in given range
Common Solution Patterns
- sin θ = sin α: θ = α + 2πn or θ = π – α + 2πn
- cos θ = cos α: θ = α + 2πn or θ = -α + 2πn
- tan θ = tan α: θ = α + πn
14. Law of Sines and Cosines
Law of Sines
a/sin A = b/sin B = c/sin C = 2R Where R is the circumradius of the triangle.
Use when: Two angles and one side (AAS/ASA) or two sides and non-included angle (SSA)
Law of Cosines
c² = a² + b² – 2ab cos C cos C = (a² + b² – c²)/(2ab)
Use when: Two sides and included angle (SAS) or three sides (SSS)
Area of Triangle
- Area = ½ab sin C
- Area = ½bc sin A
- Area = ½ac sin B
15. Applications of Trigonometry
Height and Distance Problems
- Angle of elevation: Angle above horizontal
- Angle of depression: Angle below horizontal
- Line of sight: Direct line from observer to object
Navigation Problems
- Bearing: Direction from north (clockwise)
- Course: Direction of travel
- Drift: Effect of current/wind
Periodic Motion
- Simple harmonic motion: y = A sin(ωt + φ)
- Frequency: f = ω/(2π)
- Period: T = 2π/ω
16. Problem-Solving Strategies
General Approach
- Draw a diagram when possible
- Identify given information and what to find
- Choose appropriate method (ratios, laws, identities)
- Set up equations carefully
- Solve systematically
- Check reasonableness of answers
Common Problem Types
- Right triangle problems: Use basic ratios
- Oblique triangle problems: Use laws of sines/cosines
- Identity verification: Use algebraic manipulation
- Equation solving: Use standard techniques
- Application problems: Model with trigonometric functions
17. Key Formulas Summary
Basic Ratios
- sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj
Fundamental Identity
Double Angle
- sin 2θ = 2 sin θ cos θ
- cos 2θ = cos²θ – sin²θ
Sum Formulas
- sin(A ± B) = sin A cos B ± cos A sin B
- cos(A ± B) = cos A cos B ∓ sin A sin B
Laws for Triangles
- Law of Sines: a/sin A = b/sin B = c/sin C
- Law of Cosines: c² = a² + b² – 2ab cos C
18. Common Mistakes to Avoid
- Confusing degrees and radians
- Forgetting domain restrictions
- Incorrect sign determination in quadrants
- Mixing up sum and difference formulas
- Not checking all possible solutions
- Incorrect application of laws of sines/cosines
- Forgetting to use reference angles
- Calculation errors with special angles
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