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
2 days ago - In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that angle to the length of the longest side of the ...
Reddit
reddit.com › r/mathematics › what is sine?
r/mathematics on Reddit: What is sine?
April 21, 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
44
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.
2 of 13
12
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.
Videos
43:05
Graphing the Sine & Cosine Functions - [2-21-8] - YouTube
14:07
Graphing Sine and Cosine Functions with Transformations (Multiple ...
Understand the different ways we understand the sine function - ...
The Sine Function: f(x) = sin(x) - YouTube
13:05
The Sine Function Explained - YouTube
48:06
05 - Sine and Cosine - Definition & Meaning - Part 1 - What is ...
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
Is sine function even or odd?
The sine function is an odd function whereas cosine is an even function.
byjus.com
byjus.com › maths › sine-function
Sine Function Definition
What 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
BlocksCAD
blockscad3d.com › editor
BlocksCAD
Looking to use BlocksCAD in the classroom · Our blocks-based programming interface is easy to use, so teachers and students spend less time getting started and more time on important, and aligned, content standards · BlocksCAD is a cloud-based 3D modeling tool that encourages students to ...
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 ...
Top answer 1 of 7
25
All of the values in your picture can be deduced from two theorems:
- The Pythagorean theorem: If a right triangle has sides
where 
is the hypotenuse, then 
- If a right triangle has an angle of
, then the length of the side opposite to that angle is half the length of the hypotenuse.
Both can be proven with elementary high school geometry.
Let's see how this works for Quadrant I (angles between 
and 
), as the rest follows from identities.
; this is clear from the definition.
; less intuitive because it breaks the triangle, but 
, and 
because the adjacent side and the hypotenuse coincide when the angle is 
.
; this corresponds to an isosceles triangle, and if we set the sides to be 
, then by the Pythagorean theorem, the hypotenuse is 
.
; this follows immediately from theorem 2 above.
; if we take a 
triangle, and set the side opposite to the 
angle to be 
, then the hypotenuse is 
and the side opposite to the 
angle satisfies 
, so its length is 
- and thus 
.
2 of 7
18
As in Alfonso Fernandez's answer, the remarkable values in your diagram can be calculated with basic plane geometry. Historically, the values for the trig functions were deduced from those using the half-angle and angle addition formulae. So since you know 30°, you can then use the half-angle formula to compute 15, 7.5, 3.25, 1.125, and 0.5625 degrees. Now use the angle addition formula to compute 0.5625° + 0.5625° = 2*0.5625°, and so on for 3*0.5625°, 4*0.5625°...
These would be calculated by hand over long periods of time, then printed up in long tables that filled entire books. When an engineer or a mariner needed to know a particular trig value, he would look up the closest value available in his book of trig tables, and use that.
Dominic Michaelis points out that in higher math the trig functions are defined without reference to geometry, and this allows one to come up with explicit formulae for them. You may reject this as mere formalist mumbo-jumbo, but conceptually I find that the university-level definitions for the trig functions make much more sense than the geometric ones, because it clears the mystery on why these functions turn up in situations that have nothing to do with angles or circles. So eventually you may lose your desire to have the values computed from the geometrical definition.
Of course, if you're going to be using the geometrical definition anyway, you could also just grab a ruler and a protractor and measure away all night, and compute a table of trig values that way.
One final note: you're still using the "ratio of sides of a triangle" definition for the trig functions. I strongly recommend you abandon this definition in favor of the circular definition: 
is the height of an angle 
, divided by the length of the arm of the angle, 
is the same for the width of an angle, and 
is the slope of the arm of the angle. The reason why I recommend you use this definition is because, while it's as conceptually meaningful as the triangular one (once you think about it for a second), it allows you to easily see where the values for angles greater than 90° are coming from. The triangular definition is so limited that I personally find it destructive to even bother teaching in school, I wonder if it wouldn't be easier to jump right in with the circular definition. I know it held me back for years.
Processing
processing.org › reference
Reference / Processing.org
Calculates the sine of an angle · tan() Calculates the ratio of the sine and cosine of an angle · += (add assign) Combines addition with assignment · + (addition) Adds two values or concatenates string values · -- (decrement) Substracts ...
Wolfram MathWorld
mathworld.wolfram.com › Sine.html
Sine -- from Wolfram MathWorld
March 15, 2011 - The sine function is one of the basic functions encountered in trigonometry (the others being the cosecant, cosine, cotangent, secant, and tangent). Let be an angle measured counterclockwise from the x-axis along an arc of the unit circle.
Quora
quora.com › How-is-sine-defined-as-a-function
How is sine defined as a function? - Quora
Answer (1 of 6): There are a number of ways to define sine as a function, depending on what rules you allow for the definition. One way is to say that \sin x = -i\Im e^{ix}. Some would argue that that’s shifting the problem from “how do you define sine” to “how do you define complex integration”...
College Board
apstudents.collegeboard.org › courses › ap-precalculus
AP Precalculus – AP Students | College Board
You’ll model and explore periodic phenomena using transformations of trigonometric functions. Topics may include: Relating right triangle trigonometry to the sine, cosine, and tangent functions · Modeling data and scenarios with sinusoidal functions · Using inverse trigonometric functions to solve trigonometric equations ·
Math is Fun
mathsisfun.com › sine-cosine-tangent.html
Sine, Cosine, Tangent
With these three magical functions, we're equipped to solve all sorts of real-world questions, because they let us: ... And we want to know "d" (the distance down). 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… x 30 d = 18.88 to 2 decimal places. ... Try this paper-based exercise where you can calculate the sine function for all angles from 0° to 360°, and then graph the result.
Wikipedia
en.wikipedia.org › wiki › Trigonometric_functions
Trigonometric functions - Wikipedia
1 week ago - 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. They are widely used in all sciences that are related to geometry, such as ...
Gatech
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DSP First
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Mathematics LibreTexts
math.libretexts.org › campus bookshelves › rio hondo › math 175: plane trigonometry › chapter 2: graphing trigonometric functions
2.4: Transformations Sine and Cosine Functions - Mathematics LibreTexts
March 14, 2023 - Recall that the sine and cosine functions relate real number values to the \(x\)- and \(y\)-coordinates of a point on the unit circle. So what do they look like on a graph on a coordinate plane? Let’s start with the sine function. We can create a table of values and use them to sketch a graph.
Cuemath
cuemath.com › trigonometry › sine-function
Sine - Graph, Table, Properties, Examples | Sine Function
The sine of an angle is a trigonometric function that is denoted by sin x, where x is the angle in consideration. In a right-angled triangle, the ratio of the perpendicular and the hypotenuse is called the sine function. In other words, it is the ratio of the side opposite to the angle in ...
Math Open Reference
mathopenref.com › sine.html
The sine function - math word definition - Math Open Reference
The sine function, along with cosine and tangent, is one of the three most common trigonometric functions. In any right triangle, the sine of an angle x is the length of the opposite side (O) divided by the length of the hypotenuse (H).
Godot Engine
docs.godotengine.org › en › stable › tutorials › scripting › gdscript › gdscript_basics.html
GDScript reference — Godot Engine (stable) documentation in English
GDScript is a high-level, object-oriented, imperative, and gradually typed programming language built for Godot. It uses an indentation-based syntax similar to languages like Python. Its goal is to...
Alloprof
alloprof.qc.ca › en › students › vl › mathematics › the-sine-function-m1171
The Sine Function | Secondaire | Alloprof
A sine function is a periodic function defined by the |y|-coordinates of the points on the unit circle in relation to the measure of the circle’s angles (in radians).
Math Medic
portal.mathmedic.com › lesson-plans
Math Medic Teacher Portal
Math Medic is a web application that helps teachers and students with math problems.
Galway Maths Grinds
galwaymathsgrinds.wordpress.com › maths-topics › general-form-of-sine-function
General Form of Sine Function | Galway Maths Grinds
April 7, 2021 - The general form of a sine function is: y = sin (Bx-C) + D. Where: Amplitude: = |A|. The amplitude is the magnitude of the stretch or compression of thefunction from its parent function: y = sin x. Period: = 2π/B. The period of a trigonometric ...
Washington State Standard
washingtonstatestandard.com › home › live final day coverage of the 2026 washington state legislative session
Live final day coverage of the 2026 WA legislative session • Washington State Standard
2 weeks ago - Follow along here as lawmakers in Olympia work toward adjourning sine die in the final hours of the 60-day session, on March 12, 2026.