Trigonometry Basics Quiz

Trigonometry

• Definition: Branch of mathematics dealing with the relationships between the angles and sides of triangles, particularly right triangles.

• Basic Functions:

  • Sine (sin): Ratio of the opposite side to the hypotenuse.
    • sin(θ) = Opposite/Hypotenuse
  • Cosine (cos): Ratio of the adjacent side to the hypotenuse.
    • cos(θ) = Adjacent/Hypotenuse
  • Tangent (tan): Ratio of the opposite side to the adjacent side.
    • tan(θ) = Opposite/Adjacent

• Reciprocal Functions:

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

• Pythagorean Identity:

  • sin²(θ) + cos²(θ) = 1

• Angle Measures:

  • Degrees: Full circle = 360 degrees.
  • Radians: Full circle = 2π radians; 180 degrees = π radians.

• Special Angles:

  • 0°, 30°, 45°, 60°, 90°
  • Corresponding sine and cosine 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

• Trigonometric Identities:

  • 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 Formulas:
    • sin(2θ) = 2sin(θ)cos(θ)
    • cos(2θ) = cos²(θ) - sin²(θ)

• Applications:

  • Used in physics, engineering, architecture, and computer graphics.
  • Essential for solving problems involving angles and distances.

• Graphs:

  • Sin and cos functions oscillate between -1 and 1.
  • Tan function has vertical asymptotes and repeats every π radians.

• Law of Sines:

  • a/sin(A) = b/sin(B) = c/sin(C), where a, b, c are sides opposite to angles A, B, C respectively.

• Law of Cosines:

  • c² = a² + b² - 2ab*cos(C), relating the lengths of the sides of a triangle to the cosine of one of its angles.

Trigonometry

• Definition: A mathematical branch focused on relationships between angles and sides in triangles, especially right triangles.

Basic Functions

• Sine (sin): Defines the ratio of the opposite side to the hypotenuse. Formula: sin(θ) = Opposite/Hypotenuse.
• Cosine (cos): Determines the ratio of the adjacent side to the hypotenuse. Formula: cos(θ) = Adjacent/Hypotenuse.
• Tangent (tan): Represents the ratio of the opposite side to the adjacent side. Formula: tan(θ) = Opposite/Adjacent.

Reciprocal Functions

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

Pythagorean Identity

• Key relationship: sin²(θ) + cos²(θ) = 1.

Angle Measures

• Degrees: A full circle consists of 360 degrees.
• Radians: A full circle equals 2π radians; 180 degrees corresponds to π radians.
• Special Angles: Notable angles include 0°, 30°, 45°, 60°, and 90° with corresponding sine and cosine 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.

Trigonometric Identities

• 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 Formulas:
  • sin(2θ) = 2sin(θ)cos(θ).
  • cos(2θ) = cos²(θ) - sin²(θ).

Applications

• Widely used in physics, engineering, architecture, and computer graphics, particularly in angle and distance problem-solving.

Graphs

• Sine and cosine functions oscillate between -1 and 1.
• Tangent function features vertical asymptotes and repeats every π radians.

Laws of Triangles

• Law of Sines: a/sin(A) = b/sin(B) = c/sin(C), where a, b, c are the lengths of sides opposite angles A, B, C, respectively.
• Law of Cosines: c² = a² + b² - 2ab*cos(C), linking side lengths with the cosine of an angle.

Trigonometry Overview

• Study of relationships between the angles and sides of triangles, particularly in right triangles.

Key Functions

• Sine (sin): Measures the ratio of the length of the opposite side to the hypotenuse.
• Cosine (cos): Measures the ratio of the length of the adjacent side to the hypotenuse.
• Tangent (tan): Measures the ratio of the length of the opposite side to the adjacent side.

Reciprocal Functions

• Cosecant (csc): Inverse of sine, calculated as ( \csc(\theta) = \frac{1}{\sin(\theta)} ).
• Secant (sec): Inverse of cosine, calculated as ( \sec(\theta) = \frac{1}{\cos(\theta)} ).
• Cotangent (cot): Inverse of tangent, calculated as ( \cot(\theta) = \frac{1}{\tan(\theta)} ).

Pythagorean Identity

• Fundamental identity expressing the relationship between sine and cosine: ( \sin^2(\theta) + \cos^2(\theta) = 1 ).

Angles

• Degrees and Radians:
  • Full circle: 360 degrees equals ( 2\pi ) radians.
  • Conversion formula: ( \text{Degrees} = \text{Radians} \times \frac{180}{\pi} ).

Unit Circle

• A circle with a radius of 1, centered at the origin, essential for understanding trigonometric functions.
• Coordinates on the circle provide sine and cosine values at any angle ( \theta ): ( (x, y) = (\cos(\theta), \sin(\theta)) ).

Trigonometric Identities

• Angle Sum and Difference:
  • Sine and cosine identities allow calculation of sine and cosine for combined angles.
• Double Angle Formulas:
  • Express sine and cosine for twice an angle in terms of the original angle.

Applications

• Trigonometry plays a critical role in various fields including physics, engineering, and computer graphics.
• Key for solving problems involving oscillations, waves, and rotations.

Graphing

• Sine and cosine functions oscillate between -1 and 1, demonstrating periodic behavior.
• Tangent function features vertical asymptotes at points where ( \cos(\theta) = 0 ).

Solving Trigonometric Equations

• Techniques include manipulating equations algebraically, utilizing identities, and graphing functions for solutions.

Inverse Trigonometric Functions

• Used to determine angle measures from known ratios: ( \sin^{-1}(x) ), ( \cos^{-1}(x) ), and ( \tan^{-1}(x) ).