SEC in Google Sheets: Use Reciprocal Cosine for Signed Ratios
Calculate Sheets secant with degree conversion, a hypotenuse example, symmetry checks and explicit singularity limits.
Verification status: See the evidence and test scope described in this guide. How formula examples are checked
Quick answer: SEC returns secant from a radian angle. =SEC(60*PI()/180) is expected to give 2, while the documented zero input gives 1.
Availability and version history
- Introduced: first service launch date is not established by the official history reviewed; no numbered desktop introduction is assigned.
- Availability: documented in the rolling Google Sheets service. Examples use ordinary web-spreadsheet numeric cells; other clients and imported files are not separately certified.
- Checked: October 14, 2026; official SEC reference. Documentation and independent calculations, without native Sheets execution.
Syntax and interpretation
=SEC(angle)
Google specifies a radian angle. Convert a degree source explicitly before the call. The result is the reciprocal of cosine, a ratio rather than an angle.
The official examples use direct radian inputs, including positive, negative and zero cases. Their support does not establish every magnitude bound or nonnumeric coercion; this guide does not borrow Excel-specific limits.
Example 1: compare zero and signed secants
Enter 0, 45, 60, 135, 240 degrees in A2:A6 and fill =SEC(A2*PI()/180) down. Expected values are approximately 1, 1.414213562, 2, −1.414213562, −2.
Secant can be negative for an oriented circular angle. A positive triangle-length interpretation uses an appropriately chosen acute angle instead. Keep the source model explicit before deciding whether a sign should be retained.
Example 2: compute a hypotenuse from an adjacent side
For a stated acute right triangle with adjacent side 10 and angle 40 degrees:
=10*SEC(40*PI()/180)
Expected hypotenuse: approximately 13.054072893. Multiplying the unrounded result by COS of the same radian angle reconstructs approximately 10.
Do not substitute the opposite side into this formula. Side definitions and angle units matter even when the spreadsheet returns a finite number without an error.
Advanced check: use an identity on a moderate angle
At 30 degrees, secant is approximately 1.154700538 and tangent is approximately 0.577350269. Consequently:
=SEC(PI()/6)^2-TAN(PI()/6)^2
Expected result: approximately 1, from 4/3−1/3. TAN shares cosine in its denominator with SEC.
The identity checks the intended circular function family. It does not guarantee identical native precision for all algebraic forms or every near-singular input.
Advanced example: keep symmetry separate from direction
SEC of 60 and −60 degrees gives 2 in both cases because cosine is even. The output alone therefore cannot recover which sign the angle had. At 240 degrees the expected value is −2, showing that not every different circular position has the same secant.
Keep original angles when a later calculation needs orientation. A principal inverse of cosine cannot reconstruct the original phase from one reciprocal value.
Singularities and practical limits
The mathematical ratio is undefined at pi/2 plus integer multiples of pi. Google’s short help does not pin a complete native error table for these cases. A formula using PI() may evaluate a nearby finite representation and produce a very large number rather than reveal an exact singularity.
Inspect source units, stored precision and model validity before applying a generic fallback. A large finite result is not proof that an exact vertical geometry has a meaningful finite secant. Native text coercion, extreme-angle reduction and cross-client accuracy remain outside these independent controls.
References and verification
Inputs are fictional. Approximate numerical answers are independently calculated mathematical expectations rounded to nine places, not captured native results. Complete coercion, extreme-magnitude and cross-client behavior remain untested.