Trigonometry Basics

Study Notes

Trigonometry

Definition: Study of relationships between the angles and sides of triangles, particularly right triangles.
Basic Functions:
- Sine (sin): Ratio of the opposite side to the hypotenuse.
- Cosine (cos): Ratio of the adjacent side to the hypotenuse.
- Tangent (tan): Ratio of the opposite side to the adjacent side.
Reciprocal Functions:
- Cosecant (csc): 1/sin
- Secant (sec): 1/cos
- Cotangent (cot): 1/tan
Pythagorean Identity:
- sin²(θ) + cos²(θ) = 1
Angle Measures:
- Degrees (°): Full circle = 360°
- Radians (rad): Full circle = 2π rad; Conversion: 180° = π rad
Trigonometric Ratios in Right Triangles:
- For a right triangle with angle θ:
  - Opposite side = side opposite to θ
  - Adjacent side = side next to θ
  - Hypotenuse = longest side
Unit Circle:
- A circle with a radius of 1 centered at the origin (0,0).
- Important for defining sine and cosine for all angles.
- Coordinates of points on the unit circle: (cos(θ), sin(θ))
Trigonometric Graphs:
- Sine Function: Periodic with a period of 2π; range [-1, 1].
- Cosine Function: Periodic with a period of 2π; range [-1, 1].
- Tangent Function: Periodic with a period of π; range (-∞, ∞).
Key Angles:
- 0°, 30°, 45°, 60°, 90° (and their radian equivalents)
- Common values:
  - sin(30°) = 1/2
  - sin(45°) = √2/2
  - sin(60°) = √3/2
  - cos(30°) = √3/2
  - cos(45°) = √2/2
  - cos(60°) = 1/2
Inverse Trigonometric Functions:
- Used to find angles given a ratio.
- Notations:
  - sin⁻¹, cos⁻¹, tan⁻¹
Applications:
- Used in physics, engineering, computer graphics, and architecture.
- Essential for solving problems involving angles, heights, and distances.
Laws of Sines and Cosines:
- Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)
- Law of Cosines: c² = a² + b² - 2ab*cos(C)
Trigonometric Identities:
- Co-Function Identities: sin(90° - θ) = cos(θ)
- Sum and Difference Formulas:
  - sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B)
  - cos(A ± B) = cos(A)cos(B) ∓ sin(A)sin(B)
Graphing Tips:
- Identify amplitude, period, phase shift, and vertical shift for sine and cosine graphs.
- Understand asymptotes and periodicity for tangent graphs.

Trigonometry Overview

Focuses on the relationships among triangles' angles and sides, especially in right triangles.

Basic Trigonometric Functions

Sine (sin): Opposite side to hypotenuse ratio.
Cosine (cos): Adjacent side to hypotenuse ratio.
Tangent (tan): Opposite side to adjacent side ratio.

Reciprocal Trigonometric Functions

Cosecant (csc): Inverse of sine (1/sin).
Secant (sec): Inverse of cosine (1/cos).
Cotangent (cot): Inverse of tangent (1/tan).

Pythagorean Identity

Fundamental relationship: sin²(θ) + cos²(θ) = 1.

Angle Measurement

Degrees: A full circle comprises 360°.
Radians: A full circle comprises 2π rad; 180° corresponds to π rad.

Right Triangle Trigonometric ratios

Defined sides:
- Opposite: Side opposite angle θ.
- Adjacent: Side next to angle θ.
- Hypotenuse: Longest side in a right triangle.

Unit Circle

Circle with a radius of 1 centered at (0,0); crucial for sine and cosine definitions across all angles.
Point coordinates: (cos(θ), sin(θ)).

Trigonometric Graphs

Sine Function: Period of 2π, range from -1 to 1.
Cosine Function: Period of 2π, range from -1 to 1.
Tangent Function: Period of π, ranges from -∞ to ∞.

Key Angles and Their Values

Important angles: 0°, 30°, 45°, 60°, 90°; corresponding radian values.
Common values include:
- sin(30°) = 1/2, sin(45°) = √2/2, sin(60°) = √3/2.
- cos(30°) = √3/2, cos(45°) = √2/2, cos(60°) = 1/2.

Inverse Trigonometric Functions

Enable angle determination from given ratios.
Notation includes sin⁻¹, cos⁻¹, tan⁻¹.

Applications of Trigonometry

Essential in fields like physics, engineering, computer graphics, and architecture.
Critical for solving problems related to angles, heights, and distances.

Laws of Sines and Cosines

Law of Sines: a/sin(A) = b/sin(B) = c/sin(C) relates sides and angles.
Law of Cosines: c² = a² + b² - 2ab*cos(C) aids in finding unknown lengths.

Key Trigonometric Identities

Co-Function Identities: sin(90° - θ) = cos(θ).
Sum and Difference Formulas:
- sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B).
- cos(A ± B) = cos(A)cos(B) ∓ sin(A)sin(B).

Tips for Graphing Trigonometric Functions

Identify amplitude, period, phase, and vertical shifts for sine and cosine functions.
For tangent graphs, recognize asymptotes and periodic behavior.

Description

Explore the fundamental concepts of trigonometry, including definitions, basic functions, and key identities. This quiz covers essential topics such as sine, cosine, tangent, and the unit circle, providing a solid foundation for understanding triangles and angle measures.