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.

Common Misconceptions

• Ratios vary with different angles, and the unit circle serves as a valuable visual tool for understanding these variations.

This quiz covers the fundamental concepts of trigonometry, including definitions, basic and reciprocal functions, and key trigonometric values. Test your understanding of angle measurements and the unit circle with these important concepts in geometry.