Positive & Negative Angles
Understanding Angles
Angles measure the rotation between two rays sharing a common endpoint (vertex). In trigonometry:
Angles are considered positive when measured counter-clockwise and negative when measured clockwise from the positive x-axis.
Learning Objectives
- Understand the concepts of positive and negative angles
- Convert between degrees and radians
- Visualize angle measurement on a coordinate plane
- Recognize coterminal angles (angles differing by full rotations: ±360° or ±2π radians)
- Apply angle concepts to trigonometric functions
The Unit Circle
Fundamental Concept
The unit circle is a circle with radius 1 centered at the origin (0,0) of the coordinate plane. It provides a powerful geometric representation of trigonometric functions:
Where θ is the angle formed with the positive x-axis. The unit circle helps visualize periodic nature, symmetries, and relationships between trig functions.
Learning Objectives
- Understand the geometric definition of sine and cosine
- Relate coordinates on the unit circle to trig functions
- Identify values of sine and cosine for common angles (0°, 30°, 45°, etc.)
- Recognize the periodic nature of trig functions
- Understand the relationship between angle measure and position on the circle
Angle: 0°
cos(θ): 1.00
sin(θ): 0.00
tan(θ): 0.00
Coordinates: (1.00, 0.00)
The Pythagorean Identity
Fundamental Identity
The Pythagorean Identity is one of the most important relationships in trigonometry, derived from the Pythagorean Theorem applied to the unit circle:
This identity holds for all real values of θ and demonstrates the inherent relationship between sine and cosine functions.
Learning Objectives
- Understand the derivation of the Pythagorean Identity
- Apply the identity to simplify trigonometric expressions
- Derive related identities: 1 + tan²(θ) = sec²(θ), 1 + cot²(θ) = csc²(θ)
- Use the identity to find unknown trig function values
- Recognize applications in calculus and physics
Proof
For any point (x, y) on the unit circle:
x = cos(θ), y = sin(θ)
By Pythagorean Theorem: x² + y² = 1 (radius squared)
Therefore: cos²(θ) + sin²(θ) = 1
Graphs of Trigonometric Functions
Visualizing Functions
Trigonometric functions are periodic, meaning they repeat their values in regular intervals. Key characteristics:
Sine and cosine have period 2π, tangent has period π. Understanding graphs helps analyze function behavior.
Learning Objectives
- Recognize the graphs of sin(x), cos(x), and tan(x)
- Understand periodicity and amplitude
- Identify key points: zeros, maxima, minima
- Understand phase shifts and vertical translations
- Apply transformations to trig functions
Function Characteristics
- sin(x): Period 2π, Range [-1, 1], Odd function
- cos(x): Period 2π, Range [-1, 1], Even function
- tan(x): Period π, Range (-∞, ∞), Undefined at π/2 + kπ
Trigonometric Identities
Fundamental Relationships
Trigonometric identities are equations that are true for all values of the variable. They are essential for simplifying expressions, solving equations, and proving mathematical relationships.
Learning Objectives
- Recall and apply fundamental trigonometric identities
- Simplify complex trigonometric expressions
- Prove identities using algebraic manipulation
- Solve trigonometric equations using identities
- Apply identities in calculus contexts (derivatives, integrals)
Sum & Difference
These identities express trig functions of sum/difference of angles in terms of functions of individual angles.
Tangent & Cotangent
Derived from sine and cosine identities, useful for angle addition with tangent.
Double Angle
Express functions of double angles in terms of single angles.
Triple Angle
Useful for solving equations with triple angles and in Fourier analysis.