Trigonometry in Mathematics 11th

Trigonometry in Mathematics 11th

Basic Definitions

• Trigonometric Ratios: Relationships between angles and sides in a right triangle.
  • Sine (sin) = Opposite / Hypotenuse
  • Cosine (cos) = Adjacent / Hypotenuse
  • Tangent (tan) = Opposite / Adjacent

Key Functions

• Reciprocal Functions:
  • Cosecant (csc) = 1/sin
  • Secant (sec) = 1/cos
  • Cotangent (cot) = 1/tan
• Unit Circle: Circle with a radius of 1, centered at the origin (0,0).
  • Coordinates represent (cos θ, sin θ).

Important Angles

• Common Angles: 0°, 30°, 45°, 60°, 90°
  • sin values: 0, 1/2, √2/2, √3/2, 1
  • cos values: 1, √3/2, √2/2, 1/2, 0
  • tan values: 0, √3/3, 1, √3, undefined

Trigonometric Identities

• Pythagorean Identity:
  • sin²θ + cos²θ = 1
• Angle Sum and Difference:
  • sin(α ± β) = sinα cosβ ± cosα sinβ
  • cos(α ± β) = cosα cosβ ∓ sinα sinβ
  • tan(α ± β) = (tanα ± tanβ) / (1 ∓ tanα tanβ)

Solving Trigonometric Equations

• Techniques include:
  • Algebraic manipulation
  • Using identities
  • Inverse trigonometric functions (arcsin, arccos, arctan)

Applications

• Real-World Problems:
  • Modeling periodic phenomena (e.g., sound waves, tides)
  • Navigation and surveying (calculating distances and angles)

Graphing Trigonometric Functions

• Sine and Cosine: Period = 2π, amplitude = 1
• Tangent: Period = π, asymptotes at (2n+1)π/2 where n is an integer

Angle Measurement

• Degrees vs Radians:
  • Conversion: π radians = 180°
  • Use radians for calculus and more advanced applications

Law of Sines and Law of Cosines

• Law of Sines: a/sinA = b/sinB = c/sinC
• Law of Cosines: c² = a² + b² - 2ab cosC

Key Concepts to Remember

• Understand the unit circle and angle measurements.
• Familiarity with trigonometric identities is crucial for simplifying expressions.
• Practice solving a variety of equations and real-world applications to enhance understanding.

Basic Definitions

• Trigonometric ratios define relationships in right triangles:
  • Sine (sin) ratio compares opposite side to hypotenuse.
  • Cosine (cos) ratio compares adjacent side to hypotenuse.
  • Tangent (tan) ratio compares opposite side to adjacent side.

Key Functions

• Reciprocal functions provide alternative ways to express trigonometric ratios:
  • Cosecant (csc) is the reciprocal of sine.
  • Secant (sec) is the reciprocal of cosine.
  • Cotangent (cot) is the reciprocal of tangent.
• The unit circle, with a radius of 1, helps visualize trigonometric functions.
  • Each point on the circle corresponds to (cos θ, sin θ) coordinates.

Important Angles

• Common angles and their 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

Trigonometric Identities

• Pythagorean identity emphasizes relationships between sine and cosine:
  • sin²θ + cos²θ = 1.
• Angle sum and difference formulas (used for calculating combined angles):
  • sin(α ± β) = sinα cosβ ± cosα sinβ
  • cos(α ± β) = cosα cosβ ∓ sinα sinβ
  • tan(α ± β) = (tanα ± tanβ) / (1 ∓ tanα tanβ)

Solving Trigonometric Equations

• Common techniques for solving trigonometric equations:
  • Algebraic manipulation to simplify.
  • Applying identities for transformation.
  • Utilizing inverse trigonometric functions (arcsin, arccos, arctan).

Applications

• Trigonometry applies to real-world scenarios:
  • Models periodic phenomena such as sound waves and tides.
  • Facilitates navigation and surveying by determining distances and angles.

Graphing Trigonometric Functions

• Sine and cosine functions have a period of 2π and an amplitude of 1.
• Tangent function has a period of π and features vertical asymptotes at (2n+1)π/2 where n is any integer.

Angle Measurement

• Understanding degrees and radians is essential:
  • Conversion between angles is done via 180° = π radians.
  • Radians are commonly used in calculus and advanced mathematical applications.

Law of Sines and Law of Cosines

• Law of Sines allows finding unknown sides/angles:
  • a/sinA = b/sinB = c/sinC.
• Law of Cosines is useful for finding a side given two sides and their included angle:
  • c² = a² + b² - 2ab cosC.

Key Concepts to Remember

• Master the unit circle for angle understanding and relationship visualization.
• Familiarity with identities aids in expression simplification.
• Practice various equations and real-world problems to reinforce trigonometric concepts.