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Degree

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Degree

gives the number of radians in one degree, with numerical value Pi/180 (~= 0.0174533).

Notes Multiply by Degree to convert degrees to radians, so 30 Degree is 30 degrees. It is a mathematical constant: it has attributes Constant and Protected, NumericQ\[Degree\] is True, and D\[Degree, x\] is 0. N\[Degree, prec\] evaluates it to any precision.

Examples (4)

Every input below was run against the current Mathilda build and its output recorded.

Applications (4)

In[1]:= N[Degree]
Out[1]= 0.0174533

In[2]:= Sin[30 Degree] // N
Out[2]= 0.5

In[3]:= N[Tan[60 Degree], 40]
Out[3]= 1.7320508075688772935274463415058723669427

In[4]:= N[Degree, 40]
Out[4]= 0.017453292519943295769236907684886127134428

Options & behaviour

The constant values for Pi, E, Catalan, GoldenRatio, and Degree all live in the central numeric constant table (src/numeric.c); their identities (attributes Constant, Protected) are stamped in numeric_init.

Implementation notes

Definition. Degree is the number of radians in one degree, i.e. the constant π/180 ≈ 0.0174533. Multiplying an angle in degrees by Degree converts it to radians, so 30 Degree represents 30°. It is a constant symbol: row { "Degree", M_PI/180.0, fill_mpfr_degree } of the kConstants[] table in src/numeric.c supplies its value and the numericalize_init loop stamps it Constant | Protected, so Attributes[Degree] is {Constant, Protected}. Interned name SYM_Degree (src/sym_names.c), docstring in src/info.c. D[Degree, x] -> 0 follows from Constant. Degree participates in the trigonometric closed forms through its numeric relation to Pi rather than through a rewrite rule of its own: 180 Degree == Pi returns True and Sin[90 Degree] -> 1, while the product 30 Degree itself is left as the unevaluated Times[30, Degree].

Representation & numeric value. Degree is a bare EXPR_SYMBOL. N[Degree] returns the machine double M_PI/180; N[Degree, k] runs fill_mpfr_degree, which computes mpfr_const_pi at k + 10 guard bits, divides by 180, and rounds down to the target precision. Machine value 0.0174533; N[Degree, 50] = 0.0174532925199432957692369076848861271344287188854173.

Usage & limits. Degree is the degrees-to-radians scale factor; NumericQ[Degree] is True (whitelisted in is_numeric_quantity, src/core.c). A symbolic Cos[30 Degree] is left unevaluated, but N[Cos[30 Degree]] evaluates to 0.866025. The only limit is the one shared by the Pi-derived constants: it is irrational and its digits are produced on demand by N.

  • Attributes Constant, Protected. Attributes[Degree] = {Constant, Protected}; the symbol cannot be reassigned.
  • Propagated as an exact, unevaluated symbol; NumericQ[Degree] is True and D[Degree, x] = 0.
  • Used in arguments of trigonometric functions to express angles in degrees, e.g. 30 Degree is π/6; the trig value evaluates numerically under N, e.g. N[Cos[30 Degree]] = 0.866025 (the exact symbolic form Cos[30 Degree] is left unevaluated).
  • N[Degree] gives the machine value 0.0174533; N[Degree, prec] gives any precision, e.g. N[Degree, 50] = 0.0174532925199432957692369076848861271344287188854173.

Attributes: Constant, Protected.

References

See also: N, Pi, E, Catalan, GoldenRatio

Notes & additional examples

Notes

Degree is the mathematical constant Pi/180, the number of radians in one degree. Multiply an angle by it to convert degrees to radians, so 30 Degree is the radian measure of 30 degrees and Sin[30 Degree] numericalises to 0.5. It carries the Constant and Protected attributes (D[Degree, x] is 0), is recognised by NumericQ, and evaluates to arbitrary precision under N[Degree, prec] — the last example agrees with N[Pi/180, 40] digit for digit.