Enter any angle and instantly get its secant — sec(θ) = 1/cos(θ) — plus the reciprocal cosine value, with clear "Undefined" results at every asymptote.
Enter an angle and choose degrees or radians
This sec calculator (short for secant calculator) finds the secant of any angle you enter, in either degrees or radians, and returns the result instantly. Secant is defined as sec(θ) = 1/cos(θ) — the reciprocal of cosine — and while it appears less often in introductory coursework than sine, cosine, or tangent, knowing how to find secant of 60 degrees or any other angle is a common step in precalculus, physics, and engineering problems. This tool displays the reciprocal cosine value alongside sec(θ) so the relationship is immediately visible, and it explicitly flags "Undefined" at 90°, 270°, and every other angle where cosine equals zero, instead of returning a misleading Infinity or error.
Given one angle, this calculator converts it to radians (if you entered degrees), computes cos(θ), and takes its reciprocal to get sec(θ). It also reports cos(θ) directly for reference, since secant and cosine are reciprocals of each other everywhere cosine is nonzero.
Precalculus and trigonometry students working through reciprocal-function problems, physics and engineering students computing atmospheric or optical path lengths, surveying and construction students converting horizontal to slope distances, and anyone verifying a hand calculation of 1/cos(θ) will find this tool faster and less error-prone than working the reciprocal out by hand.
Secant is one of the three "reciprocal" trig functions (alongside cosecant and cotangent) that round out the full set of six trigonometric ratios. Although less common day-to-day than cosine itself, secant appears in specific, well-established formulas — from the Mercator map projection to airmass calculations in atmospheric optics — where expressing a relationship as 1/cos (rather than cos) is the standard, historically established convention.
Cartographers use secant directly in the Mercator projection formula that maps the curved Earth onto a flat map. Astronomers and atmospheric scientists use sec(zenith angle) to estimate airmass — how much atmosphere sunlight or starlight passes through. Builders use secant to calculate rafter lengths from a roof's horizontal run and pitch angle. Surveyors convert a horizontal distance into a slope distance using secant, and physics students use it in inclined-plane and projectile-range problems.
How sec(θ) is derived from cosine, and its reciprocal relationship
Secant and cosine are reciprocals: sec(θ) = 1/cos(θ), so wherever cosine is zero, secant is undefined.
Because cosine is bounded between -1 and 1, its reciprocal secant can never fall strictly inside that range.
Secant repeats every 360° (2π radians) — the same period as cosine, since it's cosine's direct reciprocal.
From entering an angle to reading the secant result
Type your angle into the numeric input field, for example 60.
Select the correct unit using the toggle so the conversion is accurate.
The calculator converts to radians (if needed) and computes cos(θ).
At 90°, 270°, and so on, secant has no value; the calculator flags this instead of showing an error.
Both values are displayed together in the results panel for quick reference.
Finding the secant of a 60° angle
A student needs sec(60°) for a homework problem and wants to verify the calculator's result by hand.
Explanation: 60° is one of the standard reference angles where cosine takes a clean value (0.5), making its reciprocal secant equally clean (2). This makes it a convenient angle for verifying the reciprocal relationship sec(θ) × cos(θ) = 1.
Domain, range, and period of secant compared with its reciprocal cosine
| Function | Domain | Range | Period |
|---|---|---|---|
| sec(θ) | All reals except 90°, 270°, ... (90°+n180°) | |y| ≥ 1, i.e. (-∞,-1] ∪ [1,∞) | 360° (2π rad) |
| cos(θ) (reciprocal) | All real numbers | -1 ≤ y ≤ 1 | 360° (2π rad) |
Reading the result: a numeric sec(θ) value tells you 1 divided by cos(θ) at that angle; an "Undefined" result means the angle falls on one of secant's vertical asymptotes, where cosine is zero.
Typical ranges: sec(θ) never falls strictly between -1 and 1 — it jumps straight from ≥1 to ≤-1 as cosine crosses zero, and grows very large in magnitude near an asymptote.
Manual verification: compute cos(θ) yourself (a scientific calculator or unit circle table works), take its reciprocal, and confirm it matches the displayed sec(θ) — then multiply your result by the displayed cos(θ) and confirm you get 1.
Where secant genuinely shows up outside the classroom
The projection's vertical scale factor is built directly from sec(latitude).
Astronomers estimate atmospheric path length using sec(zenith angle).
Builders find rafter length as horizontal run × sec(roof pitch angle).
Convert a horizontal distance to a slope distance using secant of the elevation angle.
Estimate slant range from altitude and elevation angle using secant.
Range and force formulas on inclined planes often include a sec² term.
Signal path length through the atmosphere scales with secant of the elevation angle.
Compute a diagonal member's length from its horizontal span and angle.
Angle-of-view and off-axis calculations in optical systems use secant terms.
The secant modulus of a stress-strain curve borrows the same reciprocal-cosine naming convention.
Some distance-correction formulas on curved surfaces use secant terms.
Check reciprocal-function and unit-circle problems that call for secant specifically.
Verify a cosine table or calculator result using the reciprocal relationship.
What this sec calculator does well, and where it doesn't apply
How sec(θ) and its reciprocal cos(θ) compare at the standard reference angles
| Angle (θ) | cos(θ) | sec(θ) |
|---|---|---|
| 0° | 1 | 1 |
| 30° | √3/2 ≈ 0.8660 | 2/√3 ≈ 1.1547 |
| 45° | √2/2 ≈ 0.7071 | √2 ≈ 1.4142 |
| 60° | 0.5 | 2 |
| 90° | 0 | Undefined |
Summary: This sec calculator gives you an instant, free way to find the secant of any angle in degrees or radians, along with its reciprocal cosine value and clear "Undefined" flags at every asymptote. Pair it with related tools like the Cot Calculator and Cosec Calculator to cover all three reciprocal trig functions.
Common questions about the secant function
Trusted educational references to go deeper on secant and the trig functions
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