"In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths."
The ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle.
Trigonometric ratios: Understanding the basic trigonometric ratios (sine, cosine, and tangent) and how they relate to right triangles.
Unit circle: Learning about the unit circle and how it is used to visualize and understand the trigonometric functions.
Periodicity: Understanding the concept of periodicity and how it relates to cosine function.
Amplitude: Understanding the concept of amplitude and how it relates to the graph of the cosine function.
Angular measurement: Understanding the concepts of angles in radians and degrees and how they relate to the cosine function.
Graphing: Learning about how to graph the cosine function and understanding its key features including maximum and minimum values, zeros, period, and amplitude.
Calculus of cosine: Learning about how to differentiate and integrate the cosine function using calculus.
Inverse cosine: Understanding the concept of inverse cosine and how it is used to find angles given the cosine value.
Trigonometric identities: Learning about trigonometric identities and how they can be used to simplify expressions involving cosine function.
Applications: Understanding the various applications of cosine function in real-world contexts, including physics, engineering, and mathematics.
Basic Cosine Function: A basic cosine function is a periodic function that oscillates between 1 and -1 with a period of 2π.
Shifted Cosine Function: A shifted cosine function is a cosine function that has been moved horizontally or vertically on the coordinate plane. The horizontal shift is usually denoted by a phase shift.
Damped Cosine Function: A damped cosine function is a periodic function with a damping factor that causes the amplitude of the function to decay over time.
Inverse Cosine Function: An inverse cosine function, also known as arccosine, is the inverse of the cosine function. It returns the angle whose cosine is a given number.
Hyperbolic Cosine Function: A hyperbolic cosine function, also known as cosh, is a mathematical function related to the exponential function. It is used in physics, engineering, and other fields.
Complex Cosine Function: A complex cosine function is a complex-valued function that can be used to represent oscillating electrical signals or quantum mechanical wave functions.
"They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others."
"The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent."
"Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used."
"Each of these six trigonometric functions has a corresponding inverse function, and an analog among the hyperbolic functions."
"The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles."
"To extend the sine and cosine functions to functions whose domain is the whole real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used."
"The domain of the other functions is the real line with some isolated points removed."
"Modern definitions express trigonometric functions as infinite series or as solutions of differential equations."
"This allows extending the domain of sine and cosine functions to the whole complex plane."
"The domain of the other trigonometric functions is the complex plane with some isolated points removed."
"They are among the simplest periodic functions and are widely used for studying periodic phenomena through Fourier analysis."
"The trigonometric functions (also called circular functions, angle functions or goniometric functions)"
"such as navigation, solid mechanics, celestial mechanics, geodesy, and many others."
"...relate an angle of a right-angled triangle to ratios of two side lengths."
"geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used."
"Each of these six trigonometric functions has a corresponding inverse function, and an analog among the hyperbolic functions."
"...extending the domain of sine and cosine functions to the whole complex plane."
"...the domain of the other trigonometric functions to the complex plane with some isolated points removed."
"Modern definitions express trigonometric functions as infinite series or as solutions of differential equations."