SinDegrees

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

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Basic Examples (6)

The argument is given in degrees:

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

Calculate the sine by hand:

Verify the result:

Solve a trigonometric equation:

Solve a trigonometric inequality:

Plot over two periods:

Series expansion at 0:

Scope (47)

Numerical Evaluation (6)

Evaluate numerically:

Evaluate to high precision:

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

SinDegrees can take complex number inputs:

Evaluate SinDegrees efficiently at high precision:

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

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix SinDegrees function using MatrixFunction:

Specific Values (6)

Values of SinDegrees at fixed points:

SinDegrees has exact values at rational multiples of 30 degrees:

Values at infinity:

Simple exact values are generated automatically:

More complicated cases require explicit use of FunctionExpand:

Zeros of SinDegrees:

Extrema of SinDegrees:

Find the first positive maximum as a root of :

Substitute in the result:

Visualize the result:

Visualization (4)

Plot the SinDegrees function:

Plot over a subset of the complexes:

Plot the real part of SinDegrees:

Plot the imaginary part of SinDegrees:

Polar plot with SinDegrees:

Function Properties (13)

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

Check this with FunctionPeriod:

SinDegrees is defined for all real and complex values:

SinDegrees achieves all real values between and :

The range for complex values is the whole plane:

SinDegrees is an odd function:

SinDegrees has the mirror property :

SinDegrees is an analytic function of x:

SinDegrees is monotonic in a specific range:

SinDegrees is not injective:

SinDegrees is not surjective:

SinDegrees is neither non-negative nor non-positive:

SinDegrees has no singularities or discontinuities:

SinDegrees is neither convex nor concave:

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

TraditionalForm formatting:

Differentiation (3)

First derivative:

Higher derivatives:

Formula for the derivative:

Integration (3)

Compute the indefinite integral of SinDegrees via Integrate:

Definite integral of SinDegrees over a period is 0:

More integrals:

Series Expansions (3)

Find the Taylor expansion using Series:

Plots of the first three approximations for SinDegrees around :

Fourier series:

SinDegrees can be applied to power series:

Function Identities and Simplifications (5)

Double-angle formula using TrigExpand:

Angle sum formula:

Multiple‐angle expressions:

Recover the original expression using TrigReduce:

Convert sums to products using TrigFactor:

Convert to exponentials using TrigToExp:

Function Representations (4)

Representation through CosDegrees:

The Pythagorean identity:

Representations through CosDegrees, TanDegrees and CotDegrees:

Representation through CscDegrees:

Applications (22)

Basic Trigonomometric Applications (3)

Given , find the SinDegrees of the angle :

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

Draw a circle:

Trigonomometric Identities (7)

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

Compare with the result of direct calculation:

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

Compare this result with directly calculated SinDegrees:

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

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

Simplify trigonometric expressions:

Verify trigonometric identities:

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 :

The numerical value of :

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

Calculate the base:

Get the numerical value of the base:

Trigonomometric Equations (2)

Solve a basic trigonometric equation:

Solve trigonometric equations including other trigonometric functions:

Solve trigonometric equations with conditions:

Trigonomometric Inequalities (2)

Solve this trigonometric inequality:

Solve this trigonometric inequality including other trigonometric functions:

Advanced Applications (8)

Lissajous figure:

Equiangular (logarithmic) spiral:

Plot a sphere:

Plot a torus:

Plot waves:

Approximate the almost nowhere differentiable Riemann–Weierstrass function:

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

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

Properties & Relations (11)

Check that 1 degree is radians:

Basic parity and periodicity properties are automatically applied:

Complicated expressions containing trigonometric functions do not simplify automatically:

Another example:

Use FunctionExpand to express SinDegrees in terms of radicals:

Compositions with the inverse trigonometric functions:

Solve a trigonometric equation:

Numerically find a root of a transcendental equation:

Plot the function to check if the solution is correct:

The zeros of SinDegrees:

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

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

SinDegrees is a numeric function:

Possible Issues (1)

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

With exact input, the answer is correct:

Neat Examples (5)

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

Solve a trigonometric equation:

Add some condition to the solution:

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

Indefinite integral of :

Noncommensurate waves (quasiperiodic function):

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SinDegrees

Details

Examples

Basic Examples (6)