Trigonometry Basics

Trigonometry

• Definition: Trigonometry is a branch of mathematics that studies relationships between the angles and sides of triangles, particularly right-angled triangles.

• Key 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 Functions:

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

• Pythagorean Identity: In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

  • Formula: sin²(θ) + cos²(θ) = 1

• Unit Circle:

  • A circle with a radius of 1 centered at the origin of a coordinate plane.
  • Points on the unit circle represent the values of sin(θ) and cos(θ):
    • sin(θ) = y-coordinate
    • cos(θ) = x-coordinate

• Common Angles:

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

• Law of Sines:

  • Relates the ratios of the lengths of sides of a triangle to the sines of its angles.
  • Formula: (a/sin(A)) = (b/sin(B)) = (c/sin(C))

• Law of Cosines:

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

• Applications:

  • Problem-solving in geometry, physics (e.g., waves, oscillations), engineering, and navigation.

• Trigonometric Identities:

  • Angle Sum/Difference Formulas:

    • sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B)
    • cos(A ± B) = cos(A)cos(B) ∓ sin(A)sin(B)
    • tan(A ± B) = (tan(A) ± tan(B))/(1 ∓ tan(A)tan(B))

  • Double Angle Formulas:

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

• Graphs of Trigonometric Functions:

  • Sine and cosine functions have a periodic wave shape with a period of 2π.
  • Tangent function has a periodic shape with asymptotes and a period of π.

These notes cover fundamental concepts in trigonometry, foundational for advanced studies in mathematics and related fields.

Trigonometry Definition

• The branch of mathematics that explores the relationship between angles and sides of triangles, especially right-angled triangles.

Key Trigonometric Functions

• Sine (sin): The ratio of the opposite side to the hypotenuse of a right triangle.
• Cosine (cos): The ratio of the adjacent side to the hypotenuse of a right triangle.
• Tangent (tan): The ratio of the opposite side to the adjacent side of a right triangle.

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).

Pythagorean Identity

• In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
• Formula: sin²(θ) + cos²(θ) = 1

Unit Circle

• A circle with a radius of 1 unit centered at the origin of a coordinate plane.
• Points on the unit circle represent the values of sin(θ) and cos(θ):
  • sin(θ) = y-coordinate
  • cos(θ) = x-coordinate

Common Angles:

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

Law of Sines

• Relates the ratios of the lengths of sides of a triangle to the sines of its angles.
• Formula: (a/sin(A)) = (b/sin(B)) = (c/sin(C))

Law of Cosines

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

Applications of Trigonometry

• Solving problems in geometry, physics (e.g., waves, oscillations), engineering, and navigation.

Trigonometric Identities

• Angle Sum/Difference Formulas:

  • sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B)
  • cos(A ± B) = cos(A)cos(B) ∓ sin(A)sin(B)
  • tan(A ± B) = (tan(A) ± tan(B))/(1 ∓ tan(A)tan(B))

• Double Angle Formulas:

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

Graphs of Trigonometric Functions

• Sine and cosine functions have a periodic wave shape with a period of 2π.
• Tangent function has a periodic shape with asymptotes and a period of π.