TAN

What it does

TAN(number)

TAN(number) is the circular tangent of an angle measured in radians:

tan x  =  sin x / cos x

• Domain: all reals except the odd multiples of pi/2, where cos x = 0.
• Range: all of the extended reals — tangent is surjective onto (-infinity, +infinity) on each of its branches.
• Poles: simple poles at x = (2k+1) * pi/2 for every integer k. The function changes sign across each pole: tan(x) -> +infinity from the left of pi/2 and -infinity from the right. These are genuine poles of the real function, not branch cuts; tangent is meromorphic.
• Period: pi, not 2*pi. This is the identity most often mis-remembered, and it is why a correct implementation reduces modulo pi/2 and then selects between a tangent and a cotangent kernel.
• Parity: odd.
• Series about zero: tan x = x + x^3/3 + 2 x^5/15 + 17 x^7/315 + ..., with radius of convergence pi/2 — the distance to the nearest pole. The coefficients involve the Bernoulli numbers; A&S chapter 4 gives the closed form.
• Zeros: the integer multiples of pi, same as sine.

The identity that matters for implementation is the half-angle relation tan(x + pi/2) = -1/tan(x) — the reason a tangent kernel and its reciprocal are the same kernel with a branch selector.

Microsoft's page states one remark and it is the calling convention: the argument is in radians; convert degrees by multiplying by PI()/180 or by using RADIANS.

Arguments

The misunderstood position, again, is the units. TAN(45) is not one.

Try it

Quirks

Result and edge cases

Returns Number.

The pole is the edge that has no clean answer. No binary64 number is an odd multiple of pi/2, so no representable argument sits exactly on a pole: TAN always has a finite answer to give. What it gives near a pole is enormous and almost entirely determined by how accurately the implementation knows the distance from the argument to the true pole — see Numerical notes. A reader who expects TAN(PI()/2) to be an error or an infinity should expect neither; PI()/2 is a double that differs from the real pi/2, and the tangent of that double is a large finite number.

Errors

Microsoft's page lists no errors for TAN.

How to implement

Numerical notes

TAN inherits every argument-reduction difficulty of SIN (see that page for the three-regime scheme, Cody-Waite and Payne-Hanek) and adds one of its own: conditioning near the poles.

The conditioning statement. The relative condition number of tangent is x / (sin x cos x), which blows up as x approaches a pole. Concretely: if the reduced argument r — the distance from x to the nearest odd multiple of pi/2 — carries a relative error eps, then tan x carries a relative error of roughly eps as well, but r itself is the result of a catastrophic cancellation. For an argument near a pole, almost all the significant digits of r came from the bits of pi used in the reduction. An implementation whose stored pi is short delivers a TAN near the poles whose leading digits are noise.

This is the practical reason the trig family's substrate matters more for TAN than for SIN. Sine is bounded and locally linear at its zeros, so a reduction error shows up as a small absolute error. Tangent multiplies the same reduction error by an unbounded factor.

Do not compute TAN as SIN/COS. Two separately rounded transcendental evaluations, divided, give you both errors plus the division's; worse, near a pole the COS in the denominator is itself the badly conditioned quantity, so the composition compounds exactly the error you most wanted to control. The standard formulations (Cody & Waite; fdlibm's __kernel_tan) evaluate a single rational approximation on the reduced argument and select between t and -1/t from the parity of the reduction quotient, so the reduction is performed once and the reciprocal branch reuses it.

Testing implication. A vector suite for TAN that samples uniformly over a wide interval is testing the easy part. The informative vectors are: arguments within a few ULP of k*pi + pi/2 for a spread of k; arguments within a few ULP of k*pi; and the large-argument witnesses from the reduction literature.

Rust code

20 tan "TAN" (1, 1) [V "number"] |s, a, sv| {
    raw(s, a, ExcelRealPolicy::CIRCULAR_TRIG, k::tan::tan_kernel)
}

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.

• XC-TAN-001: build 19929 versus build 20430; 1 same-input pairs, 0 differing answers. Compared axes: excel-build. Numeric answers were compared by their recorded binary64 bits; other answers by typed value.
• XC-TAN-002: build 20131 versus build 20430; 1 same-input pairs, 0 differing answers. Compared axes: excel-build. Numeric answers were compared by their recorded binary64 bits; other answers by typed value.

Agreement on these compared inputs does not establish agreement on unmeasured inputs.

Recorded environments and comparison details.

Alternatives

Relationships

• SIN and COS — TAN = SIN/COS mathematically, though a good implementation does not compute it that way (see below).
• ATAN and ATAN2 — the inverses. ATAN returns the principal branch (-pi/2, pi/2); ATAN2 recovers the quadrant from a coordinate pair and is the function to use when the angle of a vector is wanted.
• TANH — the hyperbolic sibling, related by tan(ix) = i tanh(x). The contrast is instructive: TANH is bounded and has no real poles, so it is numerically placid where TAN is not.
• RADIANS / DEGREES / PI — the conversion apparatus.

Named variants

No named variants are recorded yet.