How this instrument works
Secant, cosecant, and cotangent are not new functions built from scratch — each is simply 1 divided by one of the three functions everyone learns first. Secant is 1 ⁄ cos(angle), cosecant is 1 ⁄ sin(angle), and cotangent is 1 ⁄ tan(angle). Flip any of the three familiar ratios upside down and the reciprocal already has a name waiting for it.
A separate sheet on this site returns sine, cosine, and tangent directly for any angle you type in — those three come straight out of the unit circle with no inversion involved. This page answers a related but distinct question: given the same angle, what are the three reciprocals? Nothing here duplicates that other page's output; secant, cosecant, and cotangent are computed by inverting sine, cosine, and tangent, not by recomputing them from scratch.
These three see less everyday use than sine, cosine, and tangent, but calculus leans on them constantly. The derivative of tan(x) is sec²(x) — a fact stated far more cleanly with secant in the mix than it would be written purely in terms of cosine. Whenever cosine or sine equals zero, the matching reciprocal has no value at all: secant has no output wherever cosine hits zero, and cosecant behaves the same way wherever sine does.
- Enter the angle into the Angle field; degrees, radians, and turns are all selectable units.
- Read sec(angle) for the secant — the reciprocal of cosine.
- Read csc(angle) for the cosecant — the reciprocal of sine.
- Read cot(angle) for the cotangent — the reciprocal of tangent.
- Try 45° to see secant and cosecant land on the identical value, since sine and cosine match there too.
Worked example — 60°, 45°, and 30°
At angle = 60°, cos(60°) = 0.5, so sec(60°) = 1 ÷ 0.5 = 2 exactly. Sine at 60° is √3⁄2 ≈ 0.8660254037844387, so csc(60°) = 1 ÷ 0.8660254037844387 ≈ 1.1547005383792517. Tangent at 60° is √3 ≈ 1.7320508075688772, so cot(60°) = 1 ÷ 1.7320508075688772 ≈ 0.5773502691896258.
At 45°, cosine and sine are both √2⁄2 ≈ 0.7071067811865476, so secant and cosecant come out identical: both ≈ 1.4142135623730951, which is √2. Tangent at 45° is exactly 1, so cotangent is exactly 1 too — the only angle in this range where a function matches its own reciprocal. At 30°, the roles from 60° trade places: sec(30°) ≈ 1.1547005383792517 and csc(30°) = 2 exactly, while cot(30°) ≈ 1.7320508075688772.
Questions
What is secant in terms of cosine?
Secant is the reciprocal of cosine: sec(angle) = 1 ÷ cos(angle). At 60°, cos(60°) = 0.5, so sec(60°) = 1 ÷ 0.5 = 2 exactly — a whole number, though most angles give a decimal.
How does cosecant differ from secant?
Cosecant inverts sine rather than cosine: csc(angle) = 1 ÷ sin(angle). At 60°, sin(60°) ≈ 0.866, giving csc(60°) ≈ 1.155 — a different figure from secant's clean 2, simply because sine and cosine themselves differ at that angle. At 45°, where sine and cosine match, secant and cosecant match too.
What is cotangent, and where does it come from?
Cotangent is the reciprocal of tangent: cot(angle) = 1 ÷ tan(angle). Because tangent itself equals sine divided by cosine, cotangent can also be written as cos(angle) ÷ sin(angle). At 45°, tangent equals 1, so cotangent equals 1 as well.
Why compute these three instead of sine, cosine, and tangent directly?
Because the reciprocals show up on their own, especially once calculus enters the picture — the derivative of tan(x) is sec²(x), not an expression built from plain cosine. A separate sheet on this site returns sine, cosine, and tangent themselves; this one inverts them, saving a division step wherever secant, cosecant, or cotangent is the quantity actually needed.
Can secant or cosecant be undefined?
Yes. Secant has no value wherever cosine equals zero — 90°, 270°, and their repeats — and cosecant has no value wherever sine equals zero, at 0°, 180°, and so on. This calculator's input range stops short of those exact limits so it never has to divide by zero.
Do secant and cosecant ever fall below 1?
Not within this calculator's 0°–90° range: cosine and sine themselves never exceed 1 there, so their reciprocals never drop under 1. Cotangent behaves differently — it climbs above 1 for angles under 45°, sits at exactly 1 at 45°, and falls under 1 past it, tracking tangent's own rise in reverse.