Trigonometry: Sine, Cosine, Tangent

Study Notes

Sine and Cosine

Definitions:
- Sine (sin): Opposite side over hypotenuse in a right triangle.
- Cosine (cos): Adjacent side over hypotenuse in a right triangle.
Unit Circle:
- Sine corresponds to the y-coordinate, cosine to the x-coordinate.
- For an angle θ:
  - sin(θ) = y
  - cos(θ) = x
Key Values:
- sin(0°) = 0, sin(90°) = 1, sin(180°) = 0
- cos(0°) = 1, cos(90°) = 0, cos(180°) = -1

Tangent and Cotangent

Definitions:
- Tangent (tan): Opposite side over adjacent side (tan(θ) = sin(θ)/cos(θ)).
- Cotangent (cot): Adjacent side over opposite side (cot(θ) = 1/tan(θ)).
Key Relationships:
- tan(θ) = sin(θ)/cos(θ)
- cot(θ) = cos(θ)/sin(θ)
Key Values:
- tan(0°) = 0, tan(90°) = undefined
- cot(0°) = undefined, cot(90°) = 0

Trigonometric Identities

Pythagorean Identities:
- sin²(θ) + cos²(θ) = 1
- 1 + tan²(θ) = sec²(θ)
- 1 + cot²(θ) = csc²(θ)
Reciprocal Identities:
- sin(θ) = 1/csc(θ)
- cos(θ) = 1/sec(θ)
- tan(θ) = 1/cot(θ)
Angle Sum and Difference:
- sin(α ± β) = sin(α)cos(β) ± cos(α)sin(β)
- cos(α ± β) = cos(α)cos(β) ∓ sin(α)sin(β)

Inverse Trigonometric Functions

Purpose: Determine angles from known trigonometric ratios.
Functions:
- arcsin (inverse of sin)
- arccos (inverse of cos)
- arctan (inverse of tan)
Range:
- arcsin: [-π/2, π/2]
- arccos: [0, π]
- arctan: (-π/2, π/2)

Applications of Trigonometry

Geometry: Finding lengths and angles in triangles.
Physics: Analyzing waves, oscillations, and circular motion.
Engineering: Structural analysis, design of mechanical components.
Astronomy: Calculating distances and angles between celestial bodies.
Navigation: Determining course angles and distances using triangulation.

Sine and Cosine

Sine (sin) is defined as the ratio of the length of the opposite side to the hypotenuse in a right triangle.
Cosine (cos) is the ratio of the length of the adjacent side to the hypotenuse in a right triangle.
In the unit circle, sine corresponds to the y-coordinate and cosine corresponds to the x-coordinate for an angle θ.
Key sine values:
- sin(0°) = 0
- sin(90°) = 1
- sin(180°) = 0
Key cosine values:
- cos(0°) = 1
- cos(90°) = 0
- cos(180°) = -1

Tangent and Cotangent

Tangent (tan) is determined as the ratio of the opposite side to the adjacent side, expressed as tan(θ) = sin(θ)/cos(θ).
Cotangent (cot) is the ratio of the adjacent side to the opposite side, given by cot(θ) = 1/tan(θ).
Relationship of tangent and cotangent:
- tan(θ) = sin(θ)/cos(θ)
- cot(θ) = cos(θ)/sin(θ)
Key tangent values:
- tan(0°) = 0
- tan(90°) is undefined
Key cotangent values:
- cot(0°) is undefined
- cot(90°) = 0

Trigonometric Identities

Pythagorean identities state that sin²(θ) + cos²(θ) = 1, which links sine and cosine.
Additional identities:
- 1 + tan²(θ) = sec²(θ)
- 1 + cot²(θ) = csc²(θ)
Reciprocal identities for sine, cosine, and tangent:
- sin(θ) = 1/csc(θ)
- cos(θ) = 1/sec(θ)
- tan(θ) = 1/cot(θ)
Angle sum and difference formulas:
- sin(α ± β) = sin(α)cos(β) ± cos(α)sin(β)
- cos(α ± β) = cos(α)cos(β) ∓ sin(α)sin(β)

Inverse Trigonometric Functions

Inverse trigonometric functions are used to determine angles when trigonometric ratios are known.
Types of inverse functions:
- arcsin: inverse of sine
- arccos: inverse of cosine
- arctan: inverse of tangent
Range of the inverse functions:
- arcsin: [-π/2, π/2]
- arccos: [0, π]
- arctan: (-π/2, π/2)

Applications of Trigonometry

In geometry, trigonometry aids in calculating unknown lengths and angles in triangles.
In physics, it is essential for understanding wave behavior, oscillations, and circular motion.
Engineering employs trigonometry for structural analysis and design of mechanical systems.
Astronomical calculations often use trigonometry to determine distances and angles between celestial objects.
Navigation utilizes trigonometric principles for calculating course angles and distances through triangulation.

Description

Test your understanding of sine, cosine, tangent, and their relationships in trigonometry. This quiz covers key definitions, unit circle concepts, and important identities. Assess your knowledge on the fundamental functions and their properties.