Trigonometry in Mathematics 11th

Study Notes

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.

Description

Test your knowledge of basic definitions, key functions, and important angles in trigonometry for the 11th grade. This quiz covers trigonometric ratios, identities, and solving equations. Perfect for students looking to reinforce their understanding of trigonometric concepts.