Trigonometry Basics Quiz

Trigonometry

• Definition: A branch of mathematics dealing with the relationships between the angles and sides of triangles, especially right 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

• Key Angles:

  • 0°, 30°, 45°, 60°, 90°
  • Corresponding sine values:
    • sin(0°) = 0
    • sin(30°) = 1/2
    • sin(45°) = √2/2
    • sin(60°) = √3/2
    • sin(90°) = 1
  • Corresponding cosine values:
    • cos(0°) = 1
    • cos(30°) = √3/2
    • cos(45°) = √2/2
    • cos(60°) = 1/2
    • cos(90°) = 0
  • Corresponding tangent values:
    • tan(0°) = 0
    • tan(30°) = 1/√3
    • tan(45°) = 1
    • tan(60°) = √3
    • tan(90°) = undefined

• Pythagorean Identity:

  • sin²(θ) + cos²(θ) = 1

• Angle Sum and Difference Formulas:

  • sin(α ± β) = sin(α)cos(β) ± cos(α)sin(β)
  • cos(α ± β) = cos(α)cos(β) ∓ sin(α)sin(β)
  • tan(α ± β) = (tan(α) ± tan(β)) / (1 ∓ tan(α)tan(β))

• Unit Circle:

  • A circle with a radius of 1, centered at the origin (0,0) in the Cartesian plane.
  • Angles measured in radians, where:
    • 360° = 2π radians
    • π/2, π, 3π/2 are key angles.

• Applications:

  • Used in physics, engineering, computer graphics, and any field involving periodic phenomena.
  • Important for solving triangles (law of sines and law of cosines).

• Graphs:

  • Sine and cosine functions oscillate between -1 and 1.
  • Period of sine and cosine is 2π.
  • Tangent function has a period of π and asymptotes at odd multiples of π/2.

• Important Properties:

  • Periodicity: Functions repeat at regular intervals.
  • Symmetry: Sine is odd (sin(-x) = -sin(x)), cosine is even (cos(-x) = cos(x)), tangent is odd (tan(-x) = -tan(x)).

Trigonometry Overview

• A mathematical discipline focused on the relationships between angles and sides of triangles, predominantly right triangles.

Basic Trigonometric 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 Trigonometric Functions

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

Key Angles and Their Values

• Notable angles: 0°, 30°, 45°, 60°, 90°.
• Sine values:
  • sin(0°) = 0
  • sin(30°) = 1/2
  • sin(45°) = √2/2
  • sin(60°) = √3/2
  • sin(90°) = 1
• Cosine values:
  • cos(0°) = 1
  • cos(30°) = √3/2
  • cos(45°) = √2/2
  • cos(60°) = 1/2
  • cos(90°) = 0
• Tangent values:
  • tan(0°) = 0
  • tan(30°) = 1/√3
  • tan(45°) = 1
  • tan(60°) = √3
  • tan(90°) = undefined

Pythagorean Identity

• The fundamental relationship described by the equation sin²(θ) + cos²(θ) = 1.

Angle Sum and Difference Formulas

• Sine addition and subtraction: sin(α ± β) = sin(α)cos(β) ± cos(α)sin(β).
• Cosine addition and subtraction: cos(α ± β) = cos(α)cos(β) ∓ sin(α)sin(β).
• Tangent addition and subtraction: tan(α ± β) = (tan(α) ± tan(β)) / (1 ∓ tan(α)tan(β)).

Unit Circle

• Defined as a circle with radius 1, centered at the origin (0, 0) in the Cartesian plane.
• Angle measurements: 360° corresponds to 2π radians.

Applications of Trigonometry

• Extensively utilized in fields like physics, engineering, computer graphics, and other areas dealing with periodic behavior.
• Essential in solving triangles through methods such as the law of sines and the law of cosines.

Graphs and Their Properties

• Sine and cosine functions oscillate within the range of -1 to 1, with a period of 2π.
• The tangent function has a period of π and is characterized by asymptotes at odd multiples of π/2.

Important Properties of Functions

• Periodicity: Trigonometric functions exhibit repetition at consistent intervals.
• Symmetry:
  • Sine is an odd function: sin(-x) = -sin(x).
  • Cosine is an even function: cos(-x) = cos(x).
  • Tangent is also an odd function: tan(-x) = -tan(x).