CosDegrees [θ]
gives the cosine of degrees.
CosDegrees
CosDegrees [θ]
gives the cosine of degrees.
Details
- CosDegrees and other trigonometric functions are studied in high-school geometry courses and are also used in many scientific disciplines.
- The argument of CosDegrees is assumed to be in degrees.
- CosDegrees is automatically evaluated when its argument is a simple rational multiple of ; for more complicated rational multiples, FunctionExpand can sometimes be used.
- CosDegrees of angle is the ratio of the adjacent side to the hypotenuse of a right triangle:
- CosDegrees is related to SinDegrees by the Pythagorean identity TemplateBox[{theta}, SinDegrees]^2+TemplateBox[{theta}, CosDegrees]^2=1.
- For certain special arguments, CosDegrees automatically evaluates to exact values.
- CosDegrees can be evaluated to arbitrary numerical precision.
- CosDegrees automatically threads over lists.
- CosDegrees can be used with Interval , CenteredInterval and Around objects.
- Mathematical function, suitable for both symbolic and numerical manipulation.
Examples
open all close allBasic Examples (6)
The argument is given in degrees:
Calculate CosDegrees of 45 degrees for a right triangle with unit sides:
Calculate the cosine 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:
CosDegrees can take complex number inputs:
Evaluate CosDegrees 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 CosDegrees function using MatrixFunction :
Specific Values (6)
Values of CosDegrees at fixed points:
CosDegrees 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 CosDegrees :
Extrema of CosDegrees :
Find a minimum of CosDegrees as the root of (dTemplateBox[{x}, CosDegrees])/(d x)=0 in the minimum's neighborhood:
Substitute in the result:
Visualize the result:
Visualization (4)
Plot the CosDegrees function:
Plot over a subset of the complexes:
Plot the real part of CosDegrees :
Plot the imaginary part of CosDegrees :
Polar plot with CosDegrees :
Function Properties (13)
CosDegrees is a periodic function with a period of degrees:
Check this with FunctionPeriod :
CosDegrees is defined for all real and complex values:
CosDegrees achieves all real values between and :
The range for complex values is the whole plane:
CosDegrees is an even function:
CosDegrees has the mirror property cos(TemplateBox[{z}, Conjugate])=TemplateBox[{{cos, (, z, )}}, Conjugate]:
CosDegrees is an analytic function of x:
CosDegrees is monotonic in a specific range:
CosDegrees is not injective:
CosDegrees is not surjective:
CosDegrees is neither non-negative nor non-positive:
CosDegrees has no singularities or discontinuities:
CosDegrees is neither convex nor concave:
It is concave for x in [-90,90]:
TraditionalForm formatting:
Differentiation (3)
First derivative:
Higher derivatives:
Formula for the ^(th) derivative:
Integration (3)
Compute the indefinite integral of CosDegrees via Integrate :
Definite integral of CosDegrees over a period is 0:
More integrals:
Series Expansions (3)
Find the Taylor expansion using Series :
Plot the first three approximations for CosDegrees around :
Fourier series:
CosDegrees 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 SinDegrees :
The Pythagorean identity:
Representations through SinDegrees , TanDegrees and CotDegrees :
Representation through SecDegrees :
Applications (21)
Basic Trigonometric Applications (3)
Given , find the CosDegrees of the angle :
Find the missing adjacent side length of a right triangle with hypotenuse 5, given the angle is 30 degrees:
Draw a circle:
Trigonometric Identities (7)
Calculate the CosDegrees value of 105 degrees using the sum and difference formulas:
Compare with the result of direct calculation:
Calculate the CosDegrees value of 15 degrees using the half-angle formula :
Calculate the product of two CosDegrees using the trigonometric product to sum formula :
Compare this result with directly calculated product of two CosDegrees instances:
Simplify trigonometric expressions:
Verify trigonometric identities:
Use the law of cosines to find the length of the side of the following triangle if the angle and the lengths of two other sides are , :
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 base angles :
Calculate the base:
Get the numerical value of the base:
Trigonometric Equations (2)
Solve a basic trigonometric equation:
Solve trigonometric equations including other trigonometric functions:
Solve trigonometric equations with conditions:
Trigonometric Inequalities (2)
Solve this trigonometric inequality:
Solve this trigonometric inequality including other trigonometric functions:
Advanced Applications (7)
Lissajous figure:
Equiangular (logarithmic) spiral:
Plot a sphere:
Plot a torus:
Plot 2D waves:
Approximate the almost nowhere differentiable Riemann–Weierstrass function:
Find a point in the 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 CosDegrees 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 CosDegrees :
FunctionExpand applied to CosDegrees generates expressions in trigonometric functions in radians:
ExpToTrig applied to the outputs of TrigToExp will generate trigonometric functions in radians:
CosDegrees is a numeric function:
Possible Issues (1)
Machine-precision input is insufficient to give 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 length of its sides:
Solve a trigonometric equation:
Add some condition on the solution:
Some arguments can be expressed as a finite sequence of nested radicals:
Indefinite integral of :
Non-commensurate waves (quasiperiodic function):
See Also
SinDegrees TanDegrees CotDegrees SecDegrees ArcCosDegrees Cos
Tech Notes
Related Guides
History
Text
Wolfram Research (2024), CosDegrees, Wolfram Language function, https://reference.wolfram.com/language/ref/CosDegrees.html.
CMS
Wolfram Language. 2024. "CosDegrees." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CosDegrees.html.
APA
Wolfram Language. (2024). CosDegrees. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CosDegrees.html
BibTeX
@misc{reference.wolfram_2025_cosdegrees, author="Wolfram Research", title="{CosDegrees}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/CosDegrees.html}", note=[Accessed: 10-January-2026]}
BibLaTeX
@online{reference.wolfram_2025_cosdegrees, organization={Wolfram Research}, title={CosDegrees}, year={2024}, url={https://reference.wolfram.com/language/ref/CosDegrees.html}, note=[Accessed: 10-January-2026]}