Sine Function

Home > Mathematics > Trigonometry > Sine Function

The ratio of the length of the opposite side to the length of the hypotenuse in a right triangle.

Degree and Radian Measures: This topic explains the two units of angular measurement used to measure angles in trigonometry.
Unit Circle: Unit circle is a circle with a radius of one unit, used to measure angles and trigonometric functions easily.
Graphs of Sine Function: This topic explains how the sine function is represented graphically, including its domain, range, and period.
Amplitude and Period of Sine Function: This topic explains the amplitude and period of sine functions and how they affect their graph.
Finding Sine Values: This topic explains how to find the sine values of different angles, whether in degrees or radians.
Sine Function Identities: This topic explains the various identities and properties of the sine function.
Applications of Sine Function: This topic explains the applications of the sine function in real-life situations, including sound and light waves.
Trigonometric Equations Involving Sine Function: This topic explains how to solve trigonometric equations that involve the sine function.
Inverse Sine Function: This topic explains the inverse sine function, also known as arcsin, and how it is used to find angles.
Law of Sines: This topic explains the law of sines, used for finding unknown angles and sides in a triangle.
Law of Cosines: This topic explains the law of cosines, used for finding an unknown side or angle in a triangle.
Trigonometric Functions of Angles: This topic explains how other trigonometric functions, such as cosine and tangent, are related to the sine function.
Trigonometric Ratios: This topic explains the different ratios of sides in a right-angled triangle and how they relate to the sine function.
Polar Coordinates and Sine Function: This topic explains how the sine function can be used to represent polar coordinates.
Hyperbolic Sine Function: This topic explains the hyperbolic sine function, which is the analog of the sine function in hyperbolic geometry.
Simple Sine Function: This is the standard sine function that is represented as y = sin(x) on the coordinate plane. It has a repeating cycle from 0 to 2π.
Amplitude-Scaled Sine Function: This sine function is represented as y = A*sin(x), where A represents the amplitude of the function. The amplitude scales the function vertically.
Period-Scaled Sine Function: This sine function is represented as y = sin(kx), where k represents the period of the function. The period scales the function horizontally.
Phase Shifted Sine Function: This sine function is represented as y = sin(x: H), where h represents the horizontal shift (or phase shift) of the function. The phase shift moves the function horizontally.
Amplitude and Phase Shifted Sine Function: This sine function is represented as y = A*sin(x: H), where A represents the amplitude and h represents the horizontal shift. It scales and shifts the function.
Damped Sine Function: This sine function is represented as y = Ae^(-bx)*sin(cx + d), where A represents the amplitude, b represents the damping coefficient, c represents the frequency, and d represents the phase shift. This function gradually loses amplitude over time.
Fourier Series of Sine Functions: The Fourier series of a function is a series of sine functions with different amplitudes, frequencies, and phase shifts that when combined, represent the original function.
Haversine Function: This sine function is used in trigonometry to calculate the great-circle distance between two points on a sphere. It is represented as haversin(x) = sin²(x/2).
Inverse Sine Function: This function, also known as the arcsine function, is the inverse of the sine function. It returns the angle whose sine is a given value. It is represented as y = arcsin(x).
Hyperbolic Sine Function: This function, also known as the sinh function, is defined as y = (e^x: E^(-x))/2. It is commonly used in mathematics and physics.
"The sine and cosine functions are trigonometric functions of an angle."
"For the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse."
"The sine and cosine functions are denoted simply as sin θ and cos θ."
"Yes, the definitions of sine and cosine can be extended to any real value in terms of the lengths of certain line segments in a unit circle."
"More modern definitions express the sine and cosine as infinite series, or as the solutions of certain differential equations."
"The sine and cosine functions are commonly used to model periodic phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year."
"They can be traced to the jyā and koṭi-jyā functions used in Indian astronomy during the Gupta period."
"The sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse)."
"The cosine is the ratio of the length of the adjacent leg to that of the hypotenuse."
"The sine function is denoted as sin θ."
"The cosine function is denoted as cos θ."
"The sine and cosine functions are commonly used to model the position and velocity of harmonic oscillators."
"More modern definitions express the sine and cosine as infinite series, allowing their extension to arbitrary positive and negative values."
"More modern definitions express the sine and cosine as the solutions of certain differential equations, allowing their extension to complex numbers."
"The sine and cosine functions are commonly used to model average temperature variations throughout the year."
"The sine and cosine functions are commonly used to model periodic phenomena such as sound and light waves."
"For the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse."
"The definitions of sine and cosine can be extended to any real value in terms of the lengths of certain line segments in a unit circle."
"More modern definitions express the sine and cosine as infinite series, or as the solutions of certain differential equations."
"They can be traced to the jyā and koṭi-jyā functions used in Indian astronomy during the Gupta period."