Trigonometric functions - Revision history

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2025-02-18T12:18:11Z

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← Older revision		Revision as of 12:18, 18 February 2025
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	{{Trigonometry}}		{{Trigonometry}}
	[[Category:Trigonometry]]		[[Category:Trigonometry]]
			<gallery caption="Trigonometric_functions">
			File:Academ_Base_of_trigonometry.svg\|Academ Base of Trigonometry
			File:TrigonometryTriangle.svg\|Trigonometry Triangle
			File:TrigFunctionDiagram.svg\|Trig Function Diagram
			File:Periodic_sine.svg\|Periodic Sine
			File:Circle-trig6.svg\|Circle Trig 6
			File:Unit_Circle_Definitions_of_Six_Trigonometric_Functions.svg\|Unit Circle Definitions of Six Trigonometric Functions
			File:trigonometric_function_quadrant_sign.svg\|Trigonometric Function Quadrant Sign
			File:Trigonometric_functions.svg\|Trigonometric Functions
			File:Unit_circle_angles_color.svg\|Unit Circle Angles Color
			File:Taylorsine.svg\|Taylor Sine
			File:Taylor_cos.gif\|Taylor Cosine
			File:Taylorreihenentwicklung_des_Kosinus.svg\|Taylor Series Expansion of Cosine
			</gallery>

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-{{png-image}}
+== Trigonometric Functions ==
-==Definition==
+Trigonometric functions are fundamental mathematical functions that relate the angles of a triangle to the lengths of its sides. They are widely used in various fields such as [[mathematics]], [[physics]], [[engineering]], and [[computer science]]. The primary trigonometric functions are the sine, cosine, and tangent, which are defined for angles in a right triangle or on the unit circle.
-In the context of the unit circle, these functions are defined for any real angle as follows:
+=== Definitions ===
-==Properties and Applications==
+The six main trigonometric functions are:
-These functions are also used to solve [[triangle]]s, particularly in [[trigonometry]], where the lengths of a triangle's sides and its angles are related. This is crucial in many areas of [[applied mathematics]], [[physics]], and [[engineering]], such as in the design of structures, the analysis of waves, and in navigation.
+* '''Sine (sin)''': In a right triangle, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse.
-==Extensions==
+=== Unit Circle ===
-There are also [[hyperbolic trigonometric functions]] which are analogs of the trigonometric functions but for a hyperbola instead of a circle. These include hyperbolic sine (sinh), hyperbolic cosine (cosh), and hyperbolic tangent (tanh), among others.
+The unit circle is a circle with a radius of one centered at the origin of the coordinate plane. It is a fundamental tool in trigonometry, allowing the extension of trigonometric functions to all real numbers. The angle in the unit circle is measured from the positive x-axis, and the coordinates of a point on the unit circle are \((\cos \theta, \sin \theta)\).
-==Complex Numbers==
+[[File:Unit_Circle_Definitions_of_Six_Trigonometric_Functions.svg|thumb|right|Unit circle definitions of trigonometric functions]]
-Trigonometric functions can also be defined for [[complex number]]s, greatly extending their applicability. This is done using [[Euler's formula]], which relates complex exponentials to trigonometric functions, providing a powerful tool in [[complex analysis]] and other areas of mathematics.
 * [[Trigonometry]]
 * [[Unit circle]]
 * [[Euler's formula]]
-[[Category:Mathematics]]
+== Gallery ==
 [[Category:Trigonometry]]

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2024-03-25T02:41:55Z

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New page

{{png-image}}
The '''trigonometric functions''' are [[mathematics|mathematical]] functions that relate the angles of a [[triangle]] to the lengths of its sides. They are fundamental in the study of [[geometry]], [[physics]], [[engineering]], and many other fields. The most commonly used trigonometric functions are sine, cosine, and tangent, often abbreviated as sin, cos, and tan, respectively. These functions are defined for [[angle]]s, which can be measured in [[radian]]s or degrees.

==Definition==
Trigonometric functions are defined using a [[right-angled triangle]] or a [[unit circle]] (a circle with a radius of one unit). For angles in a right-angled triangle, the sine of an angle is the ratio of the length of the opposite side to the length of the [[hypotenuse]], the cosine is the ratio of the length of the adjacent side to the hypotenuse, and the tangent is the ratio of the length of the opposite side to the length of the adjacent side.

In the context of the unit circle, these functions are defined for any real angle as follows:
* The sine of an angle is the y-coordinate of the point on the unit circle at that angle,
* The cosine of an angle is the x-coordinate,
* The tangent of an angle is the y-coordinate divided by the x-coordinate (when the x-coordinate is not zero).

==Properties and Applications==
Trigonometric functions have many important properties, such as periodicity and symmetry. They are periodic in that they repeat their values in regular intervals, making them especially useful in the analysis of [[wave]]s and oscillations. The sine and cosine functions, for example, have a period of \(2\pi\) radians (or 360 degrees), meaning that their values repeat every \(2\pi\) radians.

These functions are also used to solve [[triangle]]s, particularly in [[trigonometry]], where the lengths of a triangle's sides and its angles are related. This is crucial in many areas of [[applied mathematics]], [[physics]], and [[engineering]], such as in the design of structures, the analysis of waves, and in navigation.

==Extensions==
Beyond the basic trigonometric functions, there are several related functions that are useful in various contexts. These include:
* The [[secant]] (sec), which is the reciprocal of the cosine,
* The [[cosecant]] (csc), the reciprocal of the sine,
* The [[cotangent]] (cot), the reciprocal of the tangent.

There are also [[hyperbolic trigonometric functions]] which are analogs of the trigonometric functions but for a hyperbola instead of a circle. These include hyperbolic sine (sinh), hyperbolic cosine (cosh), and hyperbolic tangent (tanh), among others.

==Complex Numbers==
Trigonometric functions can also be defined for [[complex number]]s, greatly extending their applicability. This is done using [[Euler's formula]], which relates complex exponentials to trigonometric functions, providing a powerful tool in [[complex analysis]] and other areas of mathematics.

==See Also==
* [[Inverse trigonometric functions]]
* [[Trigonometry]]
* [[Unit circle]]
* [[Euler's formula]]

[[Category:Mathematics]]
[[Category:Trigonometry]]

{{math-stub}}

← Older revision		Revision as of 12:18, 18 February 2025
Line 83:		Line 83:
	{{Trigonometry}}		{{Trigonometry}}
	[[Category:Trigonometry]]		[[Category:Trigonometry]]
			<gallery caption="Trigonometric_functions">
			File:Academ_Base_of_trigonometry.svg\|Academ Base of Trigonometry
			File:TrigonometryTriangle.svg\|Trigonometry Triangle
			File:TrigFunctionDiagram.svg\|Trig Function Diagram
			File:Periodic_sine.svg\|Periodic Sine
			File:Circle-trig6.svg\|Circle Trig 6
			File:Unit_Circle_Definitions_of_Six_Trigonometric_Functions.svg\|Unit Circle Definitions of Six Trigonometric Functions
			File:trigonometric_function_quadrant_sign.svg\|Trigonometric Function Quadrant Sign
			File:Trigonometric_functions.svg\|Trigonometric Functions
			File:Unit_circle_angles_color.svg\|Unit Circle Angles Color
			File:Taylorsine.svg\|Taylor Sine
			File:Taylor_cos.gif\|Taylor Cosine
			File:Taylorreihenentwicklung_des_Kosinus.svg\|Taylor Series Expansion of Cosine
			</gallery>