Trigonometry: Sine Function

Sine

Definition: The sine function is a trigonometric function that relates the ratio of the opposite side to the hypotenuse of a right-angled triangle.

Notation: The sine function is denoted by sin(x), where x is the angle in radians or degrees.

Unit Circle: The sine function can be defined using the unit circle, where the sine of an angle is the y-coordinate of the point on the unit circle corresponding to that angle.

Properties:

• Range: The range of the sine function is [-1, 1].
• Period: The period of the sine function is 2π (or 360°).
• Odd Function: The sine function is an odd function, meaning sin(-x) = -sin(x).
• Inverse: The inverse of the sine function is denoted by arcsin(x) or sin^(-1)(x).

Identities:

• Pythagorean Identity: sin^2(x) + cos^2(x) = 1
• Sum and Difference Formulas:
  • sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
  • sin(a - b) = sin(a)cos(b) - cos(a)sin(b)

Graph: The graph of the sine function is a wave-like curve with a maximum value of 1 and a minimum value of -1.

Applications: The sine function has many applications in physics, engineering, and mathematics, including:

• Modeling periodic phenomena, such as sound waves and light waves
• Calculating distances and angles in trigonometry
• Analyzing circular motion and oscillations

Sine Function

• Relates the ratio of the opposite side to the hypotenuse of a right-angled triangle.
• Denoted by sin(x), where x is the angle in radians or degrees.

Unit Circle Definition

• Sine of an angle is the y-coordinate of the point on the unit circle corresponding to that angle.

Properties

• Range: [-1, 1]
• Period: 2π (or 360°)
• Odd function: sin(-x) = -sin(x)
• Inverse: arcsin(x) or sin^(-1)(x)

Trigonometric Identities

• Pythagorean Identity: sin^2(x) + cos^2(x) = 1
• Sum and Difference Formulas:
  • sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
  • sin(a - b) = sin(a)cos(b) - cos(a)sin(b)

Graph Characteristics

• Wave-like curve with a maximum value of 1 and a minimum value of -1

Applications

• Modeling periodic phenomena, such as sound waves and light waves
• Calculating distances and angles in trigonometry
• Analyzing circular motion and oscillations

Trigonometric Functions

Sine (sin)

• Defined as the ratio of the opposite side to the hypotenuse of a right-angled triangle
• sin(A) equals opposite side divided by hypotenuse
• Range is between -1 and 1 (inclusive)
• Period is 2π, meaning the function repeats every 2π radians
• Odd function, meaning sin(-A) is equal to -sin(A)

Cosine (cos)

• Defined as the ratio of the adjacent side to the hypotenuse of a right-angled triangle
• cos(A) equals adjacent side divided by hypotenuse
• Range is between -1 and 1 (inclusive)
• Period is 2π, meaning the function repeats every 2π radians
• Even function, meaning cos(-A) is equal to cos(A)

Relationships between Sine and Cosine

• Pythagorean identity: sin²(A) + cos²(A) equals 1
• sin(A) equals cos(π/2 - A)
• cos(A) equals sin(π/2 - A)