trigonometric functions of an angle
Factsheet
General information
General definition 
Fields of application Trigonometry, Fourier series, Mathematical analysis.
General information
General definition
Fields of application Trigonometry, Fourier series, Mathematical analysis.
Wikipedia
en.wikipedia.org › wiki › Sine_and_cosine
Sine and cosine - Wikipedia
6 days ago - Al-Khwārizmī (c. 780–850) produced tables of sines, cosines and tangents. Muhammad ibn Jābir al-Harrānī al-Battānī (853–929) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants for each degree from 1° to 90°. In the early 17th-century, the French mathematician Albert Girard published the first use of the abbreviations sin, cos, and tan; these were further promulgated by Euler (see below).
Setosa
setosa.io › ev › sine-and-cosine
Sine and Cosine explained visually
Sine and cosine — a.k.a., sin(θ) and cos(θ) — are functions revealing the shape of a right triangle.
What is sine?
Sine is best defined visually in my opinion using the unit circle. However, there is an equation but it works using angles in radians rather than degrees, and technically goes forever. sin(θ) = θ - ( θ³ / 3! ) + (θ⁵ / 5!) - (θ⁷ / 7!) + (θ⁹ / 9!) ... [Also 3! is three factorial and 3! = 1x2x3 = 6, 5! = 1x2x3x4x5 = 120, etc] To get from sin(90°) = 1, we have to first turn 90 degrees into radians. A full circle is 360 degrees, or 2π radians. So 90 degrees becomes 2π/4 = π/2 Then put it into the infinite sum: sin(90°) = π/2 - ( (π/2)³ / 3! ) + ( (π/2)⁵ / 5!) - ( (π/2)⁷ / 7!) + ( (π/2)⁹ / 9!) sin(90°) = π/2 - ( (π³/8) / 6 ) + ( (π⁵/32) / 120) - ( (π⁷/128) / 5040) + ( (π⁹/512) / 362880) ... sin(90°) = π/2 - ( π³ / 48 ) + ( π⁵ / 3480 ) - ( π⁷ / 645120) + ( π⁹ / 185794560) ... sin(90°) = π/2 - ( 31.006 / 48 ) + ( 306.020 / 3480 ) - ( 3020.293 / 645120) + ( 29809.099 / 185794560) ... sin(90°) = 1.570796 - 0.645964 + 0.079692 - 0.004682 + 0.000160 sin(90°) = 1.000002, with errors because i didn't do all infinite terms. More on reddit.com
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April 28, 2021How to distinguish sine and sign?
Sine is a much rarer word than sign. It is almost exclusively used in the context of mathematics. If you aren’t talking about math, it’s very likely someone is saying “sign.” If it’s math-related, it’s very likely to be sine. More on reddit.com
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August 20, 2022What is a sine function in trigonometry?
In trigonometry, a sine function of an angle is the ratio of the opposite side (perpendicular) and hypotenuse of a right triangle.
byjus.com
byjus.com › maths › sine-function
Sine Function Definition
What is the value of sin 0?
Sine of zero degrees angle is equal to 0.
byjus.com
byjus.com › maths › sine-function
Sine Function Definition
What is the sine function formula?
According to sine function, sine formula is given by: · Sin A = Perpendicular/Hypotenuse · where perpendicular is the opposite side of angle A.
byjus.com
byjus.com › maths › sine-function
Sine Function Definition
Videos
21:56
How are Sine, Cosine & Triangles REALLY Connected? - YouTube
02:47
Sin Cos Tan! What Are Sine, Cosine, and Tangent? (2-MINUTE MATH!)
Learn Sin, Cos, and Tan in 5 minutes
48:06
05 - Sine and Cosine - Definition & Meaning - Part 1 - What is ...
15:15
How Do We Actually Use Sine and Cosine - YouTube
00:59
A Visual Guide to the Definition and Meaning of Sine - YouTube
Math Vault
mathvault.ca › home › higher math resource hub › foundation of higher mathematics › mathematical symbols › geometry and trigonometry symbols
List of Geometry and Trigonometry Symbols | Math Vault
April 11, 2025 - Definitive list of the most common symbols in geometry and trigonometry, categorized by function into tables along with each symbol's meaning and example.
Math is Fun
mathsisfun.com › sine-cosine-tangent.html
Sine, Cosine, Tangent
Start with:sin 39° = opposite/hypotenusesin 39° = d/30Swap Sides:d/30 = sin 39° Use a calculator to find sin 39°: d/30 = 0.6293... Multiply both sides by 30:d = 0.6293…
Wikimedia
wikimedia.org › api › rest_v1 › media › math › render › svg › a6a3c9e1e99b68fbda583330226dc87136a62315 svg
{\displaystyle {\begin{aligned}\tan(\theta )&={\frac {\sin(\ ...
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GeeksforGeeks
geeksforgeeks.org › mathematics › trigonometric-symbols
Trigonometry Symbols: List of Trigonometric Symbols with Examples - GeeksforGeeks
July 23, 2025 - These trigonometric symbols are known as: Sine · Cosine · Tangent · Cosecant · Secant · Cotangent · The mathematical symbol θ is used to denote the angle. Read More, Set Theory Symbols · Basic Math Formulas · Comment · Article Tags: Article Tags: Mathematics ·
Reddit
reddit.com › r/mathematics › what is sine?
r/mathematics on Reddit: What is sine?
April 28, 2021 -
So I get that Sin, Cos and Tan are used to find angles in a triangle using the length of sides, but what’s the equation behind the function? i.e. how does sin(90) become 1? What’s the series of calculations that have to be done?
In the way that to go from 10 to 200 you multiply 10 by 20, how do you get from sin(90) to 1?
Top answer 1 of 13
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Sine is best defined visually in my opinion using the unit circle. However, there is an equation but it works using angles in radians rather than degrees, and technically goes forever. sin(θ) = θ - ( θ³ / 3! ) + (θ⁵ / 5!) - (θ⁷ / 7!) + (θ⁹ / 9!) ... [Also 3! is three factorial and 3! = 1x2x3 = 6, 5! = 1x2x3x4x5 = 120, etc] To get from sin(90°) = 1, we have to first turn 90 degrees into radians. A full circle is 360 degrees, or 2π radians. So 90 degrees becomes 2π/4 = π/2 Then put it into the infinite sum: sin(90°) = π/2 - ( (π/2)³ / 3! ) + ( (π/2)⁵ / 5!) - ( (π/2)⁷ / 7!) + ( (π/2)⁹ / 9!) sin(90°) = π/2 - ( (π³/8) / 6 ) + ( (π⁵/32) / 120) - ( (π⁷/128) / 5040) + ( (π⁹/512) / 362880) ... sin(90°) = π/2 - ( π³ / 48 ) + ( π⁵ / 3480 ) - ( π⁷ / 645120) + ( π⁹ / 185794560) ... sin(90°) = π/2 - ( 31.006 / 48 ) + ( 306.020 / 3480 ) - ( 3020.293 / 645120) + ( 29809.099 / 185794560) ... sin(90°) = 1.570796 - 0.645964 + 0.079692 - 0.004682 + 0.000160 sin(90°) = 1.000002, with errors because i didn't do all infinite terms.
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Initially when studying mathematics, functions are all pretty simple. x2 + 1 means "take x things x times and add 1". There's an easy procedure to produce the value. But once you study more math you get to functions that have much more involved definitions, some of them requiring many nontrivial steps to compute. Sine is one of those. And it doesn't even stop there. Once you get to abstract algebra, you work with functions that you don't care how, and often don't even know how to compute the value of. You might have just defined them as "some function that solves so-and-so differential equation" and as long as you can prove one exists, for your purposes, you might not ever need to figure out how to actually compute the values it takes. What I'm getting at is this: Functions are precisely fixed mappings from some domain to some range. That's it. No more, no less. Sine is one such. There are some ways to compute its exact value at certain discrete points. But for the vast majority (infinitely many) of numbers x, sin(x) has to be approximated. We do have schemes that can approximate arbitrarily well if you run them for long enough, and computers are really good at just that. But largely we don't know the exact values of sin(x) for arbitrary points x. We approximate them as necessary and that's good enough. It is easy to incorrectly ascribe other properties to functions when you learn mathematics because most functions you encounter early on share some additional properties. But for something to be a function you actually don't need to know anything about it's range, domain and the values it takes in those, as long as you know there is some range, some domain and that the same input always gives the same output. I hope in the light of this, the first paragraph of the wikipedia article on functions makes sense to you.
Wikipedia
en.wikipedia.org › wiki › Trigonometric_functions
Trigonometric functions - Wikipedia
10 hours ago - The functions of sine and versine (1 − cosine) are closely related to the jyā and koti-jyā functions used in Gupta period Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin. (See Aryabhata's sine table.) All six trigonometric functions in current use were known in Islamic mathematics by the 9th century, as was the law of sines, used in solving triangles.
BYJUS
byjus.com › maths › sine-function
Sine Function Definition
October 12, 2021 - Sine function is one of the three primary functions in trigonometry, the others being cosine, and tan functions. The sine x or sine theta can be defined as the ratio of the opposite side of a right triangle to its hypotenuse. Here, a detailed ...
Quora
quora.com › What-is-sin-in-maths
What is sin in maths? - Quora
Answer (1 of 18): The sine function \sin(x) is a measurement of how high up a point on a unit circle is when given an angle. Similarly the cosine function is a measure of how far right the point is. Mathematicians like to use radians as a measurement of angles. This is convenient since it is als...
MDN Web Docs
developer.mozilla.org › en-US › docs › Web › JavaScript › Reference › Global_Objects › Math › sin
Math.sin() - JavaScript | MDN - Mozilla
December 29, 2025 - Math.sin(-Infinity); // NaN Math.sin(-0); // -0 Math.sin(0); // 0 Math.sin(1); // 0.8414709848078965 Math.sin(Math.PI / 2); // 1 Math.sin(Infinity); // NaN
Wolfram MathWorld
mathworld.wolfram.com › Sine.html
Sine -- from Wolfram MathWorld
March 15, 2011 - The sine function sinx is one of the basic functions encountered in trigonometry (the others being the cosecant, cosine, cotangent, secant, and tangent). Let theta be an angle measured counterclockwise from the x-axis along an arc of the unit circle. Then sintheta is the vertical coordinate of the arc endpoint, as illustrated in the left figure above.
Math is Fun
mathsisfun.com › definitions › sine.html
Sine (Illustrated Math Dictionary)
In a right angled triangle, the sine of an angle is: The length of the side opposite the angle divided by the length of the hypotenuse. The abbreviation is sin · sin θ = opposite / hypotenuse · ../geometry/images/triangle-q.js?mode=sin · ...
Varsity Tutors
varsitytutors.com › home › sine
Sine
For any right triangle, $\sin(\theta)$ will always equal the length of the opposite side divided by the length of the hypotenuse.
Study.com
study.com › math › trigonometry › trigonometric functions
Sine | Definition, Function & Formula - Lesson | Study.com
April 6, 2021 - The definition of sine is the ratio ... sin in math refers to the function to measure an angle using the formula {eq}\frac{opposite}{hypotenuse} {/eq}....
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The sine function doesn't actually operate on angles, it's a function from the real numbers to the interval [-1, 1] (or from the complex numbers to the complex numbers).
However, it just so happens that it's a very useful function when the input you give it relates to angles. In particular, if you express an angle as a number in radians (in other words, on a scale where an angle of $2\pi$ corresponds to a full circle), it gives you a value that relates to the ratio of two sides of a right-angled triangle that has that angle in one corner.
If that explanation doesn't satisfy you, then you can look at it another way - if you take it that the sine function does take an angle as input and outputs a number, then the differentiability of it relates to how its output changes as you change the angle slightly. If you go far enough in calculus, you'll learn about functions whose inputs and outputs are bizarre multi-dimensional concepts, and as long as the space of bizarre multi-dimensional concepts has the right properties, you can calculate derivatives in a meaningful sense, and if you can get your head around that then differentiating a function of an angle is small fry.
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Imagine the unit circle in the usual Cartesian plane: the set of pairs $(x, y)$ where $x$ and $y$ are real numbers. The unit circle is the set of all such pairs a distance of exactly $1$ from the origin.
Imagine a point moving around the circle. As it travels around the circle, it makes an angle of $t$ radians (not degrees!) with the positive $x$-axis. From now on we call the $x$ coordinate the cosine of $t$; and the $y$ coordinate the sine of $t$.
It's as simple as that. If you only remember this one fact you can figure everything else out: the definitions of the trig functions in terms of triangles, the shape of the graphs of the functions, and everything else.
To repeat: $\cos(t)$ and $\sin(t)$ are the $x$ and $y$ coordinates, respectively, of a point on the unit circle that makes an angle of $t$ radians with the origin and the positive $x$-axis.
There's a picture here ... https://en.wikipedia.org/wiki/Unit_circle
Quora
quora.com › In-laymans-terms-what-does-sine-mean-in-mathematics
In layman's terms, what does 'sine' mean in mathematics? - Quora
It repeats forever in both directions. Its maximum value (height above the horizontal axis) is 1 and its minimum is -1. It passes through the origin. Its single argument (input variable) has units of angular measure. The shape repeats every [math]2\pi[/math] radians, or 360 degrees.
Quora
quora.com › What-is-sine-in-mathematics
What is sine in mathematics? - Quora
Answer (1 of 3): What is sine in mathematics? The sine is a function defined by the expression \sin(z)=\frac{\exp(iz)-\exp(-iz)}{2i} where \exp is the exponential function defined by \exp(z)=\sum_{k=0}^{\infty}\frac{z^k}{k!}. Other definitions have been used in the past but the above is the Mic...
Reddit
reddit.com › r/english › how to distinguish sine and sign?
r/ENGLISH on Reddit: How to distinguish sine and sign?
August 20, 2022 -
Both "sine" and "sign" are pronounced absolutely the same.
Are there ways to help resolve this ambiguity in spoken English?
Top answer 1 of 15
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Sine is a much rarer word than sign. It is almost exclusively used in the context of mathematics. If you aren’t talking about math, it’s very likely someone is saying “sign.” If it’s math-related, it’s very likely to be sine.
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Both are used in math, but if you know what they mean and how they’re used, it’s never ambiguous. “Flip the sign of this variable.” “We need sine-theta.” “Which sign gives us a useful real-world answer?” “Sine is used because we have the lengths of the hypotenuse and the opposite leg.” It’s always clear from context.
Math.net
math.net › home › trigonometry › trigonometric functions › sine
Sine - Math.net
Sine, written as sin(θ), is one of the six fundamental trigonometric functions.