Trigonometry Basics Quiz

Trigonometry

• Definition:

  • Trigonometry is the branch of mathematics dealing with the relationships between the angles and sides of triangles, particularly right triangles.

• Basic Trigonometric Ratios:

  • Sine (sin):
    • sin(θ) = Opposite side / Hypotenuse
  • Cosine (cos):
    • cos(θ) = Adjacent side / Hypotenuse
  • Tangent (tan):
    • tan(θ) = Opposite side / Adjacent side

• Reciprocal Ratios:

  • Cosecant (csc):
    • csc(θ) = 1/sin(θ) = Hypotenuse / Opposite side
  • Secant (sec):
    • sec(θ) = 1/cos(θ) = Hypotenuse / Adjacent side
  • Cotangent (cot):
    • cot(θ) = 1/tan(θ) = Adjacent side / Opposite side

• Pythagorean Identities:

  • sin²(θ) + cos²(θ) = 1
  • 1 + tan²(θ) = sec²(θ)
  • 1 + cot²(θ) = csc²(θ)

• Trigonometric Angles:

  • Common angles: 0°, 30°, 45°, 60°, 90°
  • Corresponding values:
    • sin: 0, 1/2, √2/2, √3/2, 1
    • cos: 1, √3/2, √2/2, 1/2, 0
    • tan: 0, √3/3, 1, √3, Undefined

• Graphing Trigonometric Functions:

  • Sine and Cosine:
    • Period: 2π; Amplitude: 1
  • Tangent:
    • Period: π; Asymptotes at (2n + 1)π/2 where n is an integer

• Inverse Trigonometric Functions:

  • arcsin(x), arccos(x), arctan(x)
  • These functions provide angles corresponding to given ratios.

• Applications of Trigonometry:

  • Solving triangles (finding unknown sides and angles)
  • Modeling periodic phenomena (such as sound and light waves)
  • Navigation and surveying

• Important Formulas:

  • Law of Sines:
    • a/sin(A) = b/sin(B) = c/sin(C)
  • Law of Cosines:
    • c² = a² + b² - 2ab * cos(C)

• Trigonometric Equations:

  • Basic forms: sin(θ) = x, cos(θ) = x, tan(θ) = x
  • Solutions often involve using identities or graphing.

• Tips for Studying:

  • Memorize key ratios and angle values.
  • Practice deriving and applying identities.
  • Work through example problems for different applications.

Trigonometry Overview

• Trigonometry focuses on the relationships between angles and sides in triangles, particularly right triangles.

Basic Trigonometric Ratios

• 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 Ratios

• Cosecant (csc): Reciprocal of sine; ratio of hypotenuse to opposite side.
• Secant (sec): Reciprocal of cosine; ratio of hypotenuse to adjacent side.
• Cotangent (cot): Reciprocal of tangent; ratio of adjacent side to opposite side.

Pythagorean Identities

• Fundamental relationships:
  • sin²(θ) + cos²(θ) = 1
  • 1 + tan²(θ) = sec²(θ)
  • 1 + cot²(θ) = csc²(θ)

Common Trigonometric Angles

• Notable angles and corresponding sine, cosine, and tangent values:
  • 0°: sin = 0, cos = 1, tan = 0
  • 30°: sin = 1/2, cos = √3/2, tan = √3/3
  • 45°: sin = √2/2, cos = √2/2, tan = 1
  • 60°: sin = √3/2, cos = 1/2, tan = √3
  • 90°: sin = 1, cos = 0, tan = Undefined

Graphing Trigonometric Functions

• Sine and Cosine Functions:
  • Both have a period of 2π and an amplitude of 1.
• Tangent Function:
  • Has a period of π with asymptotes at (2n + 1)π/2 where n is an integer.

Inverse Trigonometric Functions

• Includes arcsin(x), arccos(x), and arctan(x) for determining angles from known ratios.

Applications of Trigonometry

• Useful for solving triangles, determining unknown sides and angles.
• Models periodic phenomena such as sound and light waves.
• Essential for navigation and surveying tasks.

Important Formulas

• Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)
• Law of Cosines: c² = a² + b² - 2ab * cos(C)

Trigonometric Equations

• Basic formats include sin(θ) = x, cos(θ) = x, and tan(θ) = x.
• Solutions may require applying identities or graphing techniques.

Study Tips

• Memorize essential trigonometric ratios and angle values for quick recall.
• Practice deriving and utilizing identities through various problems.
• Solve example problems to strengthen understanding and application.