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### 9.2 Trigonometric/Hyperbolic Functions

The shift-`S` (`calc-sin`) [`sin`] command computes the sine of an angle or complex number. If the input is an HMS form, it is interpreted as degrees-minutes-seconds; otherwise, the input is interpreted according to the current angular mode. It is best to use Radians mode when operating on complex numbers.

Calc's “units” mechanism includes angular units like `deg`, `rad`, and `grad`. While ‘`sin(45 deg)`’ is not evaluated all the time, the `u s` (`calc-simplify-units`) command will simplify ‘`sin(45 deg)`’ by taking the sine of 45 degrees, regardless of the current angular mode. See Basic Operations on Units.

Also, the symbolic variable `pi` is not ordinarily recognized in arguments to trigonometric functions, as in ‘`sin(3 pi / 4)`’, but the `a s` (`calc-simplify`) command recognizes many such formulas when the current angular mode is Radians and Symbolic mode is enabled; this example would be replaced by ‘`sqrt(2) / 2`’. See Symbolic Mode. Beware, this simplification occurs even if you have stored a different value in the variable ‘`pi`’; this is one reason why changing built-in variables is a bad idea. Arguments of the form ‘`x`’ plus a multiple of ‘`pi/2`’ are also simplified. Calc includes similar formulas for `cos` and `tan`.

The `a s` command knows all angles which are integer multiples of ‘`pi/12`’, ‘`pi/10`’, or ‘`pi/8`’ radians. In Degrees mode, analogous simplifications occur for integer multiples of 15 or 18 degrees, and for arguments plus multiples of 90 degrees.

With the Inverse flag, `calc-sin` computes an arcsine. This is also available as the `calc-arcsin` command or `arcsin` algebraic function. The returned argument is converted to degrees, radians, or HMS notation depending on the current angular mode.

With the Hyperbolic flag, `calc-sin` computes the hyperbolic sine, also available as `calc-sinh` [`sinh`]. With the Hyperbolic and Inverse flags, it computes the hyperbolic arcsine (`calc-arcsinh`) [`arcsinh`].

The shift-`C` (`calc-cos`) [`cos`] command computes the cosine of an angle or complex number, and shift-`T` (`calc-tan`) [`tan`] computes the tangent, along with all the various inverse and hyperbolic variants of these functions.

The `f T` (`calc-arctan2`) [`arctan2`] command takes two numbers from the stack and computes the arc tangent of their ratio. The result is in the full range from -180 (exclusive) to +180 (inclusive) degrees, or the analogous range in radians. A similar result would be obtained with `/` followed by `I T`, but the value would only be in the range from -90 to +90 degrees since the division loses information about the signs of the two components, and an error might result from an explicit division by zero which `arctan2` would avoid. By (arbitrary) definition, ‘`arctan2(0,0)=0`’.

The `calc-sincos` [`sincos`] command computes the sine and cosine of a number, returning them as a vector of the form ‘`[cos, sin]`’. With the Inverse flag [`arcsincos`], this command takes a two-element vector as an argument and computes `arctan2` of the elements. (This command does not accept the Hyperbolic flag.)

The remaining trigonometric functions, `calc-sec` [`sec`], `calc-csc` [`csc`] and `calc-cot` [`cot`], are also available. With the Hyperbolic flag, these compute their hyperbolic counterparts, which are also available separately as `calc-sech` [`sech`], `calc-csch` [`csch`] and `calc-coth` [`coth`]. (These commands do not accept the Inverse flag.)