Trigonometric Functions Chapter 3

Chapter 3: Trigonometric Functions

Key Concepts

• Trigonometric Ratios:

  • Sine (sin), Cosine (cos), Tangent (tan)
  • Cosecant (csc), Secant (sec), Cotangent (cot)
  • Relationships:
    • sin(x) = 1/csc(x)
    • cos(x) = 1/sec(x)
    • tan(x) = 1/cot(x)

• Definitions:

  • For a right triangle:
    • sin(θ) = Opposite side / Hypotenuse
    • cos(θ) = Adjacent side / Hypotenuse
    • tan(θ) = Opposite side / Adjacent side

• Unit Circle:

  • Radius of the circle = 1
  • Coordinates of a point on the circle (cos(θ), sin(θ))

Key Values

• Trigonometric values for standard angles (0°, 30°, 45°, 60°, 90°):
  • 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

Important Identities

1. Pythagorean Identity:

   • sin²(θ) + cos²(θ) = 1

2. Reciprocal Identities:

   • csc(θ) = 1/sin(θ)
   • sec(θ) = 1/cos(θ)
   • cot(θ) = 1/tan(θ)

3. Quotient Identities:

   • tan(θ) = sin(θ)/cos(θ)
   • cot(θ) = cos(θ)/sin(θ)

Trigonometric Equations

• Solve equations involving trigonometric functions using identities.
• Example: sin²(x) + sin(x) - 1 = 0 can be rewritten and factored.

Applications

• Used in various fields such as physics, engineering, and architecture.
• Useful for solving triangles and modeling periodic phenomena.

Graphs of Trigonometric Functions

• Sine and Cosine:

  • Period: 2π
  • Amplitude: 1
  • Graphs oscillate between -1 and 1.

• Tangent:

  • Period: π
  • Asymptotes at odd multiples of π/2.

Angle Measures

• Degrees to Radians:
  • θ radians = (θ × π) / 180
• Common conversions: 90° = π/2, 180° = π, 360° = 2π

Summary

• Understanding trigonometric functions involves knowing their definitions, identities, angles, and applications.
• Familiarity with graphs and solving trigonometric equations is crucial for mastering the concepts in this chapter.

Trigonometric Ratios

• Six basic trigonometric ratios: sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot).
• Reciprocal relationships:
  • sin(x) is the reciprocal of csc(x)
  • cos(x) is the reciprocal of sec(x)
  • tan(x) is the reciprocal of cot(x)

Definitions of Trigonometric Ratios

• Defined using the sides of a right triangle:
  • sin(θ) = Opposite side / Hypotenuse
  • cos(θ) = Adjacent side / Hypotenuse
  • tan(θ) = Opposite side / Adjacent side

The Unit Circle

• A circle with a radius of 1.
• Coordinates of a point on the circle are represented by (cos(θ), sin(θ)).
• Useful for visualizing trigonometric functions.

Key Trigonometric Values

• Standard angles: 0°, 30°, 45°, 60°, 90°
• Memorize the sin, cos, and tan values for these angles.

Trigonometric Identities

• Pythagorean Identity: sin²(θ) + cos²(θ) = 1
• Reciprocal Identities:
  • csc(θ) = 1/sin(θ)
  • sec(θ) = 1/cos(θ)
  • cot(θ) = 1/tan(θ)
• Quotient Identities:
  • tan(θ) = sin(θ)/cos(θ)
  • cot(θ) = cos(θ)/sin(θ)

Solving Trigonometric Equations

• Use trigonometric identities to simplify and solve equations involving trigonometric functions.

Applications of Trigonometric Functions

• Used in various fields like physics, engineering, and architecture.
• Help in solving triangles and modeling periodic phenomena.

Graphs of Trigonometric Functions

• Sine and Cosine:
  • Periodic functions with a period of 2π.
  • Amplitude of 1, oscillating between -1 and 1.
• Tangent:
  • Period of π.
  • Has asymptotes at odd multiples of π/2.

Angle Measures

• Relationship between degrees and radians:
  • θ radians = (θ × π) / 180
• Common conversions: 90° = π/2, 180° = π, 360° = 2π.

Summary

• Understanding trigonometric functions involves knowing their definitions, identities, angles, and applications.
• Familiarize yourself with the graphs of trigonometric functions and how to solve trigonometric equations.

Test your understanding of the key concepts of trigonometric functions, including ratios, definitions, and identities. This quiz will cover important values for standard angles and relationships among sine, cosine, and tangent. Challenge yourself to apply these concepts through various questions.