SinDegrees

✖
`SinDegrees`

New in 14.1

✖

SinDegrees[θ]

gives the sine of degrees.

Details

SinDegrees and other trigonometric functions are studied in high-school geometry courses and are also used in many scientific disciplines.
The argument of SinDegrees is assumed to be in degrees.
SinDegrees is automatically evaluated when its argument is a simple rational multiple of ; for more complicated rational multiples, FunctionExpand can sometimes be used.
SinDegrees of angle is the ratio of the opposite side to the hypotenuse of a right triangle:

SinDegrees is related to CosDegrees by the Pythagorean identity $TemplateBox[{theta}, SinDegrees]^2+TemplateBox[{theta}, CosDegrees]^2=1$ .
For certain special arguments, SinDegrees automatically evaluates to exact values.
SinDegrees can be evaluated to arbitrary numerical precision.
SinDegrees automatically threads over lists.
SinDegrees can be used with Interval, CenteredInterval and Around objects.
Mathematical function, suitable for both symbolic and numerical manipulation.

Examples

open allclose all

Basic Examples (6)Summary of the most common use cases

The argument is given in degrees:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-zvnbzt

Out[1]=1

Calculate SinDegrees of 45 degrees for a right triangle with unit sides:

Calculate the sine by hand:

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-8uoawe

Out[2]=2

Verify the result:

In[3]:=3

✖

https://wolfram.com/xid/0pupqdhpkk-yqnf8d

Out[3]=3

Solve a trigonometric equation:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-0yb4bb

Out[1]=1

Solve a trigonometric inequality:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-m2b8vb

Out[1]=1

Plot over two periods:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-x15

Out[1]=1

Series expansion at 0:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-8xz705

Out[1]=1

Scope (47)Survey of the scope of standard use cases

Numerical Evaluation (6)

Evaluate numerically:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-l274ju

Out[1]=1

Evaluate to high precision:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-b0wt9

Out[1]=1

The precision of the output tracks the precision of the input:

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-xth5g

Out[2]=2

SinDegrees can take complex number inputs:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-hfml09

Out[1]=1

Evaluate SinDegrees efficiently at high precision:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-di5gcr

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-bq2c6r

Out[2]=2

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-k9j849

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-dymc0w

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0pupqdhpkk-lmyeh7

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/0pupqdhpkk-d918ed

Out[4]=4

Or compute average-case statistical intervals using Around:

In[5]:=5

✖

https://wolfram.com/xid/0pupqdhpkk-cw18bq

Out[5]=5

Compute the elementwise values of an array:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-2g

Out[1]=1

Or compute the matrix SinDegrees function using MatrixFunction:

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-o5jpo

Out[2]=2

Specific Values (6)

Values of SinDegrees at fixed points:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-nww7l

Out[1]=1

SinDegrees has exact values at rational multiples of 30 degrees:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-cqeu9w

Out[1]=1

Values at infinity:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-bdij6w

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-drqkdo

Out[2]=2

Simple exact values are generated automatically:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-v87

Out[1]=1

More complicated cases require explicit use of FunctionExpand:

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-mnn

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0pupqdhpkk-p3j

Out[3]=3

Zeros of SinDegrees:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-cw39qs

Out[1]=1

Extrema of SinDegrees:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-kb0ip3

Out[1]=1

Find the first positive maximum as a root of :

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-0hynlo

Out[2]=2

Substitute in the result:

In[3]:=3

✖

https://wolfram.com/xid/0pupqdhpkk-f2hrld

Out[3]=3

Visualize the result:

In[4]:=4

✖

https://wolfram.com/xid/0pupqdhpkk-cj5txq

Out[4]=4

Visualization (4)

Plot the SinDegrees function:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-ecj8m7

Out[1]=1

Plot over a subset of the complexes:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-bomgjz

Out[1]=1

Plot the real part of SinDegrees:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-bo5grg

Out[1]=1

Plot the imaginary part of SinDegrees:

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-q61pi

Out[2]=2

Polar plot with SinDegrees:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-epb4bn

Out[1]=1

Function Properties (13)

SinDegrees is a periodic function with a period of 360 degrees:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-zf82es

Out[1]=1

Check this with FunctionPeriod:

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-ocqrm8

Out[2]=2

SinDegrees is defined for all real and complex values:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-cl7ele

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-de3irc

Out[2]=2

SinDegrees achieves all real values between and :

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-evf2yr

Out[1]=1

The range for complex values is the whole plane:

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-fphbrc

Out[2]=2

SinDegrees is an odd function:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-dnla5q

Out[1]=1

SinDegrees has the mirror property :

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-heoddu

Out[1]=1

SinDegrees is an analytic function of x:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-h5x4l2

Out[1]=1

SinDegrees is monotonic in a specific range:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-g6kynf

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-nlz7s

Out[2]=2

SinDegrees is not injective:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-gi38d7

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-ctca0g

Out[2]=2

SinDegrees is not surjective:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-hkqec4

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-hdm869

Out[2]=2

SinDegrees is neither non-negative nor non-positive:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-84dui

Out[1]=1

SinDegrees has no singularities or discontinuities:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-mdtl3h

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-mn5jws

Out[2]=2

SinDegrees is neither convex nor concave:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-kdss3

Out[1]=1

SinDegrees is concave for x in [0,180]:

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-io426y

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0pupqdhpkk-bb47uv

Out[3]=3

TraditionalForm formatting:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-6k0d4

Differentiation (3)

First derivative:

In[3]:=3

✖

https://wolfram.com/xid/0pupqdhpkk-mmas49

Out[3]=3

Higher derivatives:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-nfbe0l

Out[1]=1

Formula for the derivative:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-56llx

Out[1]=1

Integration (3)

Compute the indefinite integral of SinDegrees via Integrate:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-bponid

Out[1]=1

Definite integral of SinDegrees over a period is 0:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-b9jw7l

Out[1]=1

More integrals:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-4nbst

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-c8i6nd

Out[2]=2

Series Expansions (3)

Find the Taylor expansion using Series:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-ewr1h8

Out[1]=1

Plots of the first three approximations for SinDegrees around :

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-binhar

Out[3]=3

Fourier series:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-f64drv

Out[1]=1

SinDegrees can be applied to power series:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-f7dy9o

Out[1]=1

Function Identities and Simplifications (5)

Double-angle formula using TrigExpand:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-mjplp7

Out[1]=1

Angle sum formula:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-nfe4y

Out[1]=1

Multiple‐angle expressions:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-kgd4gx

Out[1]=1

Recover the original expression using TrigReduce:

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-ti7f8

Out[2]=2

Convert sums to products using TrigFactor:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-j5kup

Out[1]=1

Convert to exponentials using TrigToExp:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-u1m

Out[1]=1

Function Representations (4)

Representation through CosDegrees:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-df304y

Out[1]=1

The Pythagorean identity:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-uopc8v

Out[1]=1

Representations through CosDegrees, TanDegrees and CotDegrees:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-wdje4e

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-yo2yqd

Out[2]=2

Representation through CscDegrees:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-3iqpk4

Out[1]=1

Applications (22)Sample problems that can be solved with this function

Basic Trigonomometric Applications (3)

Given , find the SinDegrees of the angle :

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-pzrtpu

Out[1]=1

Find the missing opposite side length of a right triangle with hypotenuse 5 given the angle is 30 degrees:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-ull6kl

Out[1]=1

Draw a circle:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-q0x

Out[1]=1

Trigonomometric Identities (7)

Calculate the SinDegrees value of 105 degrees using the sum and difference formulas:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-uvn91a

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-22bxr8

Out[2]=2

Compare with the result of direct calculation:

In[3]:=3

✖

https://wolfram.com/xid/0pupqdhpkk-lbx8rc

Out[3]=3

Calculate the SinDegrees value of 15 degrees using the half-angle formula :

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-rj4ct4

Out[1]=1

Compare this result with directly calculated SinDegrees:

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-w17g1e

Out[2]=2

Calculate the product of two SinDegrees using the trigonometric product to sum formula :

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-xthfig

Out[1]=1

Compare this result with directly calculated product of two SinDegrees instances:

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-87co9l

Out[2]=2

Simplify trigonometric expressions:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-zp5yby

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-ci2htt

Out[2]=2

Verify trigonometric identities:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-fnpfz4

Out[1]=1

Use the law of sines to find the length of the side opposite to the angle angle, given the length of the side and :

This could be calculated via the formula :

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-0gu9is

Out[2]=2

The numerical value of :

In[3]:=3

✖

https://wolfram.com/xid/0pupqdhpkk-2rbc0y

Out[3]=3

Calculate the base length of an isosceles triangle given the leg length and the vertex angle :

Calculate the base:

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-g47lx

Out[2]=2

Get the numerical value of the base:

In[3]:=3

✖

https://wolfram.com/xid/0pupqdhpkk-ivjwir

Out[3]=3

Trigonomometric Equations (2)

Solve a basic trigonometric equation:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-9jfx4g

Out[1]=1

Solve trigonometric equations including other trigonometric functions:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-qmswx7

Out[1]=1

Solve trigonometric equations with conditions:

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-7wne7o

Out[2]=2

Trigonomometric Inequalities (2)

Solve this trigonometric inequality:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-x9jee2

Out[1]=1

Solve this trigonometric inequality including other trigonometric functions:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-pup7o

Out[1]=1

Advanced Applications (8)

Lissajous figure:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-hhm

Out[1]=1

Equiangular (logarithmic) spiral:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-v1r5y3

Out[1]=1

Plot a sphere:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-n96

Out[1]=1

Plot a torus:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-js6

Out[1]=1

Plot waves:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-j8l

Out[1]=1

Approximate the almost nowhere differentiable Riemann–Weierstrass function:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-g5b9ii

Out[1]=1

Intensity of the Fraunhofer diffraction pattern of a circular aperture versus diffraction angle:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-dvkhak

Out[1]=1

Find a point on a unit circle using CosDegrees and SinDegrees functions:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-u80xp

Out[1]=1

Properties & Relations (11)Properties of the function, and connections to other functions

Check that 1 degree is radians:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-qtrjkp

Out[1]=1

Basic parity and periodicity properties are automatically applied:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-mz1

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-ted

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0pupqdhpkk-vc2

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/0pupqdhpkk-54

Out[4]=4

Complicated expressions containing trigonometric functions do not simplify automatically:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-vtf

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-vt3

Out[2]=2

Another example:

In[3]:=3

✖

https://wolfram.com/xid/0pupqdhpkk-j62

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/0pupqdhpkk-wgk

Out[4]=4

Use FunctionExpand to express SinDegrees in terms of radicals:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-sub

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-g37

Out[2]=2

Compositions with the inverse trigonometric functions:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-qcr

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-n6i

Out[2]=2

Solve a trigonometric equation:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-ipj

Out[1]=1

Numerically find a root of a transcendental equation:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-fwo

Out[1]=1

Plot the function to check if the solution is correct:

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-ds23ya

Out[2]=2

The zeros of SinDegrees:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-kku

Out[1]=1

FunctionExpand applied to SinDegrees generates expressions in trigonometric functions in radians:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-fujuf5

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-roe9wv

Out[2]=2

ExpToTrig applied to the outputs of TrigToExp will generate trigonometric functions in radians:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-bl9bv4

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-lc9q8e

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0pupqdhpkk-egxn3i

Out[3]=3

SinDegrees is a numeric function:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-qwm

Out[1]=1

Possible Issues (1)Common pitfalls and unexpected behavior

Machine-precision input is insufficient to get a correct answer:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-wnf

Out[1]=1

With exact input, the answer is correct:

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-fza

Out[2]=2

Neat Examples (5)Surprising or curious use cases

Trigonometric functions are ratios that relate the angle measures of a right triangle to the lengths of its sides:

In[4]:=4

✖

https://wolfram.com/xid/0pupqdhpkk-2yke9s

In[6]:=6

✖

https://wolfram.com/xid/0pupqdhpkk-nytfuw

Solve a trigonometric equation:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-cvyldi

Out[1]=1

Add some condition to the solution:

In[2]:=2

✖

https://wolfram.com/xid/0pupqdhpkk-r4zle9

Out[2]=2

Some arguments can be expressed as a finite sequence of nested radicals:

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-eag

Out[1]=1

Indefinite integral of :

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-kk5

Out[1]=1

Noncommensurate waves (quasiperiodic function):

In[1]:=1

✖

https://wolfram.com/xid/0pupqdhpkk-snh

Out[1]=1

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