Trigonometric Functions on the Unit Circle

This page delves into the definitions of cosine and sine using the unit circle, as well as introducing the concept of radians.

Key points:

• Cosine $x-coordinate$ and sine $y-coordinate$ are defined using points on the unit circle.
• One radian is the angle subtended by an arc length equal to the radius.
• The relationship between degrees and radians is given: π radians = 180°.

Definition: Circonferenza goniometrica seno e coseno refers to the representation of sine and cosine on the unit circle.

Formula: sin²θ + cos²θ = 1 (Pythagorean identity)

The page also provides key values for sine and cosine at notable angles (0°, 90°, 180°, 270°, 360°).

Example: cos 0° = 1, sin 0° = 0; cos 90° = 0, sin 90° = 1

Quadrants and Tangent Function

This section explores the signs of trigonometric functions in different quadrants and introduces the tangent function.

Key points:

• The signs of sine and cosine vary depending on the quadrant.
• Tangent is defined as the ratio of sine to cosine: tan θ = sin θ / cos θ.

Definition: Circonferenza goniometrica tangente refers to the tangent function in relation to the unit circle.

The page includes diagrams showing the signs of sine and cosine in each quadrant, as well as the graphical representation of the tangent function on the unit circle.

Formula: cos θ = ±1 / √$1 + tan²θ$, sin θ = ±tan θ / √$1 + tan²θ$

Graphical Representations

This page presents the graphs of cosine, sine, and tangent functions.

Key features:

• Cosine and sine graphs have a period of 2π.
• Tangent graph has a period of π.
• Cosine graph is shifted π/2 to the left compared to the sine graph.

Highlight: Grafico coseno, seno, coseno, tangente formule are visually represented, showing their periodic nature and key characteristics.

The graphs include important points such as x-intercepts, y-intercepts, and asymptotes (for tangent).

Example: The sine function has y-intercepts at 0°, 180°, and 360°, while reaching maximum values of 1 at 90° and minimum values of -1 at 270°.

Trigonometric Values Table

This section provides a comprehensive table of trigonometric values for common angles.

The table includes:

• Angles in degrees and radians
• Sine, cosine, and tangent values

Highlight: The Circonferenza goniometrica tabella offers a quick reference for important trigonometric values.

Key angles covered: 0°, 30°, 45°, 60°, 90°, 180°, 270°, 360°

Example: For 45°, sin 45° = cos 45° = √2/2, and tan 45° = 1

The page also emphasizes the periodicity of these functions and their relationships.

Associated Angles

This section introduces the concept of associated angles and their relationships in trigonometry.

Key relationships covered:

1. Supplementary angles: (π - θ)
2. Opposite angles: (-θ)
3. Complementary angles: (π/2 - θ)

Definition: Angoli associati goniometria refers to angles that have specific relationships with a given angle θ.

For each relationship, the guide provides formulas for sine, cosine, and tangent.

Formula: For supplementary angles, cos(π - θ) = -cos θ, sin(π - θ) = sin θ, tan(π - θ) = -tan θ

Example: cos 120° = -cos(180° - 60°) = -cos 60° = -1/2

The page includes visual representations of these relationships on the unit circle.

More Associated Angles

This page continues the discussion of associated angles, focusing on angles that differ by π.

Key relationship covered:

• Angles differing by π: (π + θ)

Formula: cos(π + θ) = -cos θ, sin(π + θ) = -sin θ, tan(π + θ) = tan θ

The guide provides examples and visual representations on the unit circle.

Example: sin 210° = sin(180° + 30°) = -sin 30° = -1/2

The page also revisits opposite angles (-θ) with additional examples.

Highlight: Understanding Angoli associati formule is crucial for solving complex trigonometric problems.

Complementary Angles and Cotangent

This section explores the relationship between complementary angles and introduces the cotangent function.

Key points:

• Complementary angles sum to π/2 (90°).
• The sine of an angle equals the cosine of its complement.

Definition: Cotangent is defined as the reciprocal of tangent: cot θ = 1 / tan θ = cos θ / sin θ

The page includes a visual representation of complementary angles on the unit circle and formulas relating sine and cosine of complementary angles.

Formula: sin(π/2 - θ) = cos θ, cos(π/2 - θ) = sin θ

A table of cotangent values for common angles is provided, along with its graph.

Highlight: The Circonferenza goniometrica valori for cotangent complement those of sine, cosine, and tangent.

Inverse Trigonometric Functions

This page introduces inverse trigonometric functions, focusing on arcsine and arccosine.

Key points:

• Inverse functions "undo" the original trigonometric functions.
• The domain and range of inverse functions are restricted to ensure they are functions.

For arcsine:

• Domain: [-1, 1]
• Range: [-π/2, π/2]

For arccosine:

• Domain: [-1, 1]
• Range: [0, π]

Highlight: Graphs of inverse trigonometric functions are obtained by reflecting the original function over the line y = x and restricting the domain.

The page includes graphical representations of both arcsine and arccosine functions.

Example: arcsin(1/2) = π/6 (30°), as sin(π/6) = 1/2

Inverse Trigonometric Functions (Continued)

This page continues the discussion of inverse trigonometric functions, focusing on arccosine.

Key points:

• The arccosine function is the inverse of the cosine function with a restricted domain.
• The graph of arccosine is obtained by reflecting the cosine graph over y = x and restricting the domain.

Definition: Arccosine (arccos x) gives the angle whose cosine is x, restricted to the range [0, π].

The page includes a detailed graph of the arccosine function, showing its domain and range.

Example: arccos(1/2) = π/3 (60°), as cos(π/3) = 1/2

Highlight: Understanding inverse trigonometric functions is crucial for solving equations involving trigonometric functions.

The guide emphasizes the importance of domain and range restrictions in defining these inverse functions.

Inverse Trigonometric Functions - Part 2

Completion of inverse function analysis.

Definition: Arccosine is the inverse function of cosine when properly restricted.

Example: The domain of arccosine is [-1,1] and its range is [0,π].