Geometry: Triangles and Angles

Triangles

• Definition: A triangle is a polygon with three sides and three angles.
• Types:
  • Right triangle: one angle is 90° (π/2 radians)
  • Oblique triangle: no angle is 90°
• Properties:
  • Angle sum: 180° (π radians)
  • Side lengths: a, b, c (opposite angles A, B, C)
  • Pythagorean theorem: a² + b² = c² (for right triangles)

Angles

• Measurement units: degrees (°), radians (rad)
• Conversions:
  • 1 rad = 180/π °
  • 1 ° = π/180 rad
• Angle types:
  • Acute: 0° < θ < 90°
  • Right: θ = 90°
  • Obtuse: 90° < θ < 180°
  • Straight: θ = 180°

Waves

• Periodic functions:
  • Sine (sin), cosine (cos), and tangent (tan)
  • Period: 2π (360°)
• Wave properties:
  • Amplitude: maximum displacement from mean position
  • Frequency: number of cycles per unit time
  • Wavelength: distance between successive peaks/troughs
• Graphs:
  • Sinusoidal curves: sin(x), cos(x), tan(x)

Identities

• Fundamental identities:
  • Pythagorean identity: sin²(x) + cos²(x) = 1
  • 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)
• Double-angle and half-angle formulas:
  • sin(2x) = 2sin(x)cos(x)
  • cos(2x) = cos²(x) - sin²(x)

Applications

• Physics and engineering:
  • Motion, forces, and energies
  • Simple harmonic motion
• Navigation and geography:
  • Triangulation for location and distance
  • Latitude and longitude calculations
• Computer graphics and game development:
  • 3D transformations and projections
  • Animation and simulation

Triangles

• A triangle has three sides and three angles.
• There are two main types of triangles: right triangles with one 90° angle and oblique triangles with no 90° angle.
• The angle sum of a triangle is always 180° or π radians.
• The side lengths of a triangle are denoted by a, b, and c, opposite angles A, B, and C respectively.
• The Pythagorean theorem is a² + b² = c², applicable to right triangles.

Angles

• Angles can be measured in degrees or radians, where 1 radian is equivalent to 180/π degrees.
• One degree is equivalent to π/180 radians.
• Angles can be classified into four types: acute (0° < θ < 90°), right (θ = 90°), obtuse (90° < θ < 180°), and straight (θ = 180°).

Waves

• Periodic functions include sine, cosine, and tangent functions with a period of 2π or 360°.
• Wave properties include amplitude (maximum displacement from the mean position), frequency (number of cycles per unit time), and wavelength (distance between successive peaks or troughs).
• Graphs of sine, cosine, and tangent functions are sinusoidal curves.

Identities

• The Pythagorean identity is sin²(x) + cos²(x) = 1.
• The sum and difference formulas are sin(a+b) = sin(a)cos(b) + cos(a)sin(b) and cos(a+b) = cos(a)cos(b) - sin(a)sin(b) respectively.
• Double-angle formulas include sin(2x) = 2sin(x)cos(x) and cos(2x) = cos²(x) - sin²(x).
• Half-angle formulas exist but are not specified.

Applications

• Triangles are used in physics and engineering to calculate motion, forces, and energies, as well as in simple harmonic motion.
• Triangulation is used in navigation and geography to determine location and distance, and to calculate latitude and longitude.
• Triangles are used in computer graphics and game development for 3D transformations and projections, animation, and simulation.