Trigonometry Basics

Trigonometry

Definition:
- Study of relationships between the angles and sides of triangles.
Basic Functions:
- Sine (sin): Opposite side / Hypotenuse
- Cosine (cos): Adjacent side / Hypotenuse
- Tangent (tan): Opposite side / Adjacent side
Reciprocal Functions:
- Cosecant (csc): 1/sin
- Secant (sec): 1/cos
- Cotangent (cot): 1/tan
Pythagorean Identity:
- sin²(θ) + cos²(θ) = 1
Angle Measurements:
- Degrees: Full circle = 360°
- Radians: Full circle = 2π radians
Key Trigonometric Values:
- sin(0) = 0, sin(30) = 1/2, sin(45) = √2/2, sin(60) = √3/2, sin(90) = 1
- cos(0) = 1, cos(30) = √3/2, cos(45) = √2/2, cos(60) = 1/2, cos(90) = 0
- tan(0) = 0, tan(30) = 1/√3, tan(45) = 1, tan(60) = √3, tan(90) = undefined
Unit Circle:
- Circle with radius 1 centered at the origin (0,0).
- Coordinates of points correspond to (cos(θ), sin(θ)).
Trigonometric Identities:
- Angle Sum and Difference:
  - sin(a ± b) = sin(a)cos(b) ± cos(a)sin(b)
  - cos(a ± b) = cos(a)cos(b) ∓ sin(a)sin(b)
- Double Angle:
  - sin(2θ) = 2sin(θ)cos(θ)
  - cos(2θ) = cos²(θ) - sin²(θ)
Applications:
- Used in physics (waves, oscillations), engineering (forces, structures), and computer graphics (modeling rotations).
Graphing:
- Sine and Cosine Functions: Period = 2π, Amplitude = 1
- Tangent Function: Period = π, Asymptotes at odd multiples of π/2
Inverse Functions:
- arcsin, arccos, arctan: Used to find angles when sides are known.
Law of Sines:
- a/sin(A) = b/sin(B) = c/sin(C) for any triangle with sides a, b, c opposite to angles A, B, C.
Law of Cosines:
- c² = a² + b² - 2ab*cos(C) for any triangle.
Common Misconceptions:
- Remember that the ratios change with different angles; the unit circle helps visualize these changes.

Definition and Functions of Trigonometry

Trigonometry examines the relationships between the angles and sides of triangles.
Basic functions include:
- Sine (sin): Ratio of the length of the opposite side to the hypotenuse.
- Cosine (cos): Ratio of the length of the adjacent side to the hypotenuse.
- Tangent (tan): Ratio of the length of the opposite side to the adjacent side.

Reciprocal and Trigonometric Identities

Reciprocal functions:
- Cosecant (csc): Reciprocal of sine (1/sin).
- Secant (sec): Reciprocal of cosine (1/cos).
- Cotangent (cot): Reciprocal of tangent (1/tan).
Pythagorean identity states that sin²(θ) + cos²(θ) = 1.

Angle Measurements

Angles can be measured in degrees or radians:
- Full circle = 360° (degrees) or 2π radians.

Key Trigonometric Values

Key values of sine:
- sin(0) = 0, sin(30°) = 1/2, sin(45°) = √2/2, sin(60°) = √3/2, sin(90°) = 1.
Key values of cosine:
- cos(0) = 1, cos(30°) = √3/2, cos(45°) = √2/2, cos(60°) = 1/2, cos(90°) = 0.
Key values of tangent:
- tan(0) = 0, tan(30°) = 1/√3, tan(45°) = 1, tan(60°) = √3, tan(90°) is undefined.

Unit Circle

A circle of radius 1 centered at the origin (0,0).
Points on the circle correspond to (cos(θ), sin(θ)).

Advanced Trigonometric Identities

Angle sum and difference identities:
- sin(a ± b) = sin(a)cos(b) ± cos(a)sin(b)
- cos(a ± b) = cos(a)cos(b) ∓ sin(a)sin(b)
Double angle formulas:
- sin(2θ) = 2sin(θ)cos(θ)
- cos(2θ) = cos²(θ) - sin²(θ)

Applications of Trigonometry

Essential in various fields:
- Physics: Analyzing waves and oscillations.
- Engineering: Understanding forces and structural design.
- Computer graphics: Modeling rotations and animations.

Graphing Functions

Sine and cosine functions have a period of 2π and an amplitude of 1.
Tangent function has a period of π and features asymptotes at odd multiples of π/2.

Inverse Functions and Triangle Laws

Inverse functions include arcsin, arccos, arctan for calculating angles from known side lengths.
Law of Sines: a/sin(A) = b/sin(B) = c/sin(C) relates sides to angles in any triangle.
Law of Cosines: c² = a² + b² - 2ab*cos(C) provides relationships among the sides and angles in a triangle.