IMCOS

What it does

IMCOS(inumber)

IMCOS(inumber) returns the complex cosine. Real and imaginary coefficients use radians where a circular angle is involved. For pure imaginary bi, the mathematical result is the real cosh(b).

Try it

=IMCOS("i") gives real complex text about "1.54308063481524": its mathematical value is cosh(1). By contrast, the cosine of the real input 1 is about 0.5403. An imaginary coordinate changes circular cosine into hyperbolic growth.

Quirks

Complex inputs use lowercase i or j, and the output is text. The current parser gives #NUM! for malformed complex text and the wrapper gives #VALUE! for logical values.

The current sine/cosine helper rejects a real coefficient with magnitude at least 134217728, through its circular leaf. Its imaginary exponential can overflow before the half-sum is formed. Real-only input, a large imaginary component and an ordinary non-real pair therefore test different parts of the route.

How to implement

Parse the two coefficients and suffix before doing arithmetic; use the common complex formatter afterward. Retain suffix consistency, suppress a zero component and omit the coefficient one before the imaginary unit. Formatting and parsing have their own fifteen-significant-digit rules, so testing only the algebra misses observable behaviour.

For z=a+bi, the mathematical component formula is cos(a)*cosh(b) - i*sin(a)*sinh(b). The current sine/cosine route forms exp(b) using the worksheet exponential and a stored reciprocal for exp(-b), then forms the half-sum and half-difference. Independent sinh/cosh calls need not share that rounding history. The formula is a derivation aid; the named runtime leaves and stores in the current source define its implementation candidate. Validate text output as well as numeric components, and keep unmeasured rounding stages open.

Rust code

36 imcos "IMCOS" (1, 1) [V "inumber"] |s, a, sv| { unary_complex(s, a, |z| Ok(z.cos())) }

No implementation has been published with a Handbook test suite for this function.

Version differences

The recorded environments below distinguish a reading from a same-input comparison. Counts describe captured inputs in each environment; overlapping captures and comparisons are not totals of unique worksheet calls.

The record contains no same-input comparison between these environments. Reading different inputs on different builds does not establish either a version change or cross-build agreement.

Recorded environments and comparison details.

Alternatives

IMEXP expresses the exponential relation; IMSIN is a related circular function; COMPLEX constructs the input.

Named variants

No named variants are recorded yet.