Trigonometric Functions Overview

Study Notes

Trigonometric Functions

Definition: Trigonometric functions relate the angles of a triangle to the lengths of its sides.
Primary Functions:
1. Sine (sin):
  - Definition: sin(θ) = Opposite side / Hypotenuse
  - Range: [-1, 1]
2. Cosine (cos):
  - Definition: cos(θ) = Adjacent side / Hypotenuse
  - Range: [-1, 1]
3. Tangent (tan):
  - Definition: tan(θ) = Opposite side / Adjacent side = sin(θ) / cos(θ)
  - Range: All real numbers
Reciprocal Functions:
1. Cosecant (csc):
  - Definition: csc(θ) = 1/sin(θ)
2. Secant (sec):
  - Definition: sec(θ) = 1/cos(θ)
3. Cotangent (cot):
  - Definition: cot(θ) = 1/tan(θ) = cos(θ) / sin(θ)
Unit Circle:
- Trigonometric functions can be defined using the unit circle:
  - Coordinates of a point on the circle correspond to (cos(θ), sin(θ)).
  - The angle θ is measured from the positive x-axis.
Key Angles (in Degrees):
- 0°: sin(0) = 0, cos(0) = 1, tan(0) = 0
- 30°: sin(30) = 1/2, cos(30) = √3/2, tan(30) = √3/3
- 45°: sin(45) = √2/2, cos(45) = √2/2, tan(45) = 1
- 60°: sin(60) = √3/2, cos(60) = 1/2, tan(60) = √3
- 90°: sin(90) = 1, cos(90) = 0, tan(90) = undefined
Periodic Properties:
- Sine and Cosine functions are periodic with a period of 2π.
- Tangent function is periodic with a period of π.
Graph Characteristics:
- Sine: Starts at (0,0), oscillates between -1 and 1.
- Cosine: Starts at (0,1), oscillates between -1 and 1.
- Tangent: Passes through the origin, has vertical asymptotes.
Transformations:
- Vertical shifts, horizontal shifts, reflections, and changes in amplitude can transform trigonometric functions.
Applications:
- Used in physics for wave motion, in engineering for oscillations, in computer graphics for modeling periodic phenomena.

Trigonometric Functions Overview

Trigonometric functions connect angles of a triangle to the lengths of its sides.

Primary Functions

Sine (sin):
- Represents the ratio of the opposite side to the hypotenuse.
- Outputs values in the range of [-1, 1].
Cosine (cos):
- Defines the ratio of the adjacent side to the hypotenuse.
- Also ranges from [-1, 1].
Tangent (tan):
- Ratio of the opposite side to the adjacent side, equivalent to sin(θ) / cos(θ).
- Its range includes all real numbers.

Reciprocal Functions

Cosecant (csc):
- Defined as the reciprocal of sine: csc(θ) = 1/sin(θ).
Secant (sec):
- Defined as the reciprocal of cosine: sec(θ) = 1/cos(θ).
Cotangent (cot):
- The reciprocal of tangent: cot(θ) = 1/tan(θ), or cos(θ) / sin(θ).

Unit Circle

Trigonometric functions can be interpreted through the unit circle, with points corresponding to (cos(θ), sin(θ)).
Angles are measured counterclockwise from the positive x-axis.

Key Angles (in Degrees)

0°: sin(0) = 0, cos(0) = 1, tan(0) = 0.
30°: sin(30) = 1/2, cos(30) = √3/2, tan(30) = √3/3.
45°: sin(45) = √2/2, cos(45) = √2/2, tan(45) = 1.
60°: sin(60) = √3/2, cos(60) = 1/2, tan(60) = √3.
90°: sin(90) = 1, cos(90) = 0, tan(90) = undefined.

Periodic Properties

Sine and Cosine functions repeat every 2π (approximately 6.28).
Tangent function has a shorter period of π (approximately 3.14).

Graph Characteristics

Sine: Begins at (0,0) and oscillates between -1 and 1.
Cosine: Starts at (0,1) and also oscillates between -1 and 1.
Tangent: Passes through the origin and has vertical asymptotes (values where the function is undefined).

Transformations

Trigonometric functions can be altered by vertical shifts, horizontal shifts, reflections, and amplitude changes.

Applications

Widely used in physics to describe wave motion.
Important in engineering for representing oscillations.
Utilized in computer graphics for modeling periodic phenomena.

Description

This quiz covers the fundamental concepts of trigonometric functions, including definitions and ranges of sine, cosine, and tangent. Test your knowledge and understanding of how these functions relate to triangle geometry. Perfect for students learning about trigonometry!