Calculate the reciprocal of cosine for an angle entered in degrees.
Complete the required fields and select the calculate button.
Secant is undefined wherever cosine is zero, including 90° plus any multiple of 180°.
This secant calculator evaluates sec θ for an angle entered in degrees. Secant is the reciprocal of cosine. It appears in trigonometric identities, coordinate geometry, calculus, and right-triangle work.
The page reports a decimal rounded to six places. At angles where cosine is zero, secant is undefined and the calculation is rejected rather than displaying an enormous floating-point approximation.
For an angle supplied in radians, convert it first with degrees = radians × 180/π. Values separated by 360° are coterminal and produce the same secant.
sec θ = 1 ÷ cos θ
For conversion, radians = degrees × π/180. Secant is undefined whenever cos θ = 0, which occurs at θ = 90° + 180°k for any integer k. Its period is 360°, the same as cosine.
| Symbol or result | Meaning |
|---|---|
| θ | The entered angle in degrees. |
| cos θ | The cosine of the angle and denominator of secant. |
| sec θ | The reciprocal of cosine, when cosine is nonzero. |
| k | Any integer used to describe all undefined angles. |
| Displayed value | The decimal result rounded to six places. |
Enter 60°. Cosine of 60° is 0.5, so sec 60° = 1 ÷ 0.5 = 2. The calculator displays 2 after rounding.
At 120°, cosine is −0.5, so secant is −2. At 90°, cosine is zero and secant is undefined; the page returns a validation-style error instead of a finite result.
Secant is positive where cosine is positive, in quadrants I and IV, and negative in quadrants II and III. Its absolute value is always at least 1 for real angles where it is defined, because real cosine stays between −1 and 1.
Near 90° or 270°, cosine approaches zero and secant grows rapidly in magnitude. A small input or rounding change near those vertical asymptotes can create a large result change. Keep sufficient angle precision, and do not mistake a very large finite value near an undefined angle for the value exactly at that angle.
In a right triangle, secant of an acute angle can be read as hypotenuse divided by adjacent side, the reciprocal of adjacent over hypotenuse. This triangle picture is useful for angles between 0° and 90°, while the unit-circle definition extends secant to other quadrants and negative angles.
Some calculators label the inverse secant as sec⁻¹, but that notation can be confused with the reciprocal. On this page, “sec” always means 1/cos; it does not return an angle. If solving for an angle, use an inverse function with its stated domain and principal range.
Periodic checks are easy: adding or subtracting 360° should leave the result unchanged. Also, sec 0° = 1 and sec 180° = −1. These landmark values help catch degree-versus-radian mistakes before a result is used in a longer expression.
The reciprocal identity also gives sec²θ = 1 + tan²θ wherever both sides are defined. This offers another check when tangent is already known. Because squaring removes the sign, use the angle’s quadrant to choose positive or negative secant afterward. In calculus, secant frequently appears with tangent in derivatives and integrals, making sign and domain checks especially useful for every later calculation.
It uses degrees. Convert radians by multiplying by 180 divided by pi before entering the angle.
Cosine of 90 degrees is zero, and secant would require division by zero. The same applies at 90 plus any multiple of 180 degrees.
No. Secant is one divided by cosine. Inverse cosine returns an angle whose cosine is a supplied value.
Cosine becomes very close to zero near that angle. Taking its reciprocal magnifies small denominator changes.
No for real defined angles. Since cosine magnitude is at most one, the reciprocal’s magnitude is at least one.
Yes. Cosine and secant are even functions, so sec of negative theta equals sec of theta.
Multiply the displayed secant by cosine of the same degree angle. Away from undefined points, the product should be about one, allowing for rounding.