Trigonometry: Sine Function

Study Notes

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)