Secant Calculator

What secant means on the unit circle

Cosine and secant read the same rotation in two different ways. Put an angle on the unit circle and the cosine is the x coordinate of the point where the angle lands: at 60 degrees that point is (0.5, 0.866). Secant takes that x coordinate and divides 1 by it, so sec 60 is 1 divided by 0.5, which is 2. Nothing about the angle changed. Only the reading did.

The name comes from the Latin word for cutting. In the standard construction, extend the angle's ray past the circle until it meets the vertical line x = 1. That line touches the circle at (1, 0), so a line crossing it is a cutting line, a secant. The distance from the origin out to that crossing point is |sec theta|. At 60 degrees the crossing sits at (1, 1.732) and the ray length is 2, as the first figure shows.

Two consequences follow straight from the definition. Because a cosine cannot sit between -1 and 1, a secant can never sit between -1 and 1 either: every real secant is at least 1 or at most -1. And because cosine is negative in the second and third quadrants, secant is negative there too. The sign of sec always matches the sign of cos, and the size is always the reciprocal.

The NIST Digital Library of Mathematical Functions, section 4.14 lists secant with the other five circular functions and treats it as 1 over cosine, which is the definition this calculator uses. OpenStax Precalculus, section 5.3 builds the same function from a right triangle and adds the sign rules for the four quadrants.

The reciprocal identity behind the button

In a right triangle, cosine is adjacent over hypotenuse, so secant flips it to hypotenuse over adjacent. A ramp whose horizontal run is 4.0 ft and whose sloping surface is 5.0 ft has sec of the base angle equal to 5.0 ÷ 4.0 = 1.25. The same number comes out of the angle: arccos(4.0 ÷ 5.0) = 36.87 degrees, and 1 ÷ cos 36.87 = 1.25.

The Pythagorean identity ties secant to tangent. Divide sin squared plus cos squared by cos squared and you get tan squared plus 1 equals sec squared. At 30 degrees, tan 30 is 0.5774 and 0.5774 squared is 0.3333, so 1 + 0.3333 = 1.3333. Sec 30 is 1.1547 and 1.1547 squared is also 1.3333. The identity holds to the last digit the calculator shows.

Calculus keeps the same thread. The derivative of sec x is sec x times tan x, which is 2 × 1.732 = 3.464 at 60 degrees. The integral of sec x is the natural log of the absolute value of sec x plus tan x, a result that looks strange until you differentiate it back. Neither fact changes the arithmetic on this page, but both show that secant is a first class function rather than a label for a division.

Secant values at the angles people look up

Standard angles have exact secants, and the rest come from a cosine table or a calculator. The table below repeats the cosine next to each secant so you can check any row by hand: divide 1 by the cosine column and you get the secant column.

Angles that are not on the list are just as easy. At 40 degrees the cosine is 0.7660, so 1 ÷ 0.7660 = 1.3054. At 210 degrees the cosine is -0.8660, so 1 ÷ -0.8660 = -1.1547. The second figure plots the whole curve, and the vertical gaps are the angles where no value exists.

Where secant is undefined and why the number explodes

Cosine is zero at 90 degrees plus any whole number of half turns: 90, 270, 450, -90, and so on. Written in radians the same set is pi/2 plus k times pi. At those angles the division has no answer, and no finite value can stand in for it. A calculator that shows a huge number near 90 degrees is reporting the approach, not the value at 90.

The approach is steep. At 89.9 degrees the cosine is 0.001745, so secant is 572.96. Move a tenth of a degree closer and the secant is about ten times larger. That is why a surveying spreadsheet can return a ten digit number for what was meant to be a right angle, and why the calculator here prints undefined instead of a large number when the input lands exactly on the pole.

The negative branch matters in practice. A ray that is a hair past vertical has a negative cosine, so its secant is a large negative number, not a large positive one. Sign errors at this boundary are common in code that compares a computed secant with a threshold instead of comparing the cosine with zero.

Worked examples with real measurements

A roof surface from its run and pitch. A shed roof has a horizontal run of 12 ft and a pitch of 30 degrees above horizontal. The sloping surface is the hypotenuse of that triangle, so its length is the run times the secant of the pitch: 12 × 1.1547 = 13.86 ft. Ordering 14 ft of panel per rafter covers the extra, and the same multiplier works for any run at that pitch.

Sunlight through the atmosphere. When the sun sits 60 degrees from straight overhead, light crosses twice the thickness of air it would cross from directly above, because 1 ÷ cos 60 = 2. At 80 degrees from overhead the factor is 1 ÷ 0.1736 = 5.76. That is the plane parallel estimate of air mass. It overstates the path near the horizon, where the curved atmosphere means light never crosses that much air, but the secant factor is the right first correction for solar panel output and for color temperature at low sun.

A triangle without the angle. A brace is 25 in long and meets a rail 20 in from the corner. Secant of the angle at the corner is 25 ÷ 20 = 1.25, and the angle itself is arccos(20 ÷ 25) = 36.87 degrees. A carpenter who only owns a tape measure gets both numbers from the side ratio, which is the mode this page offers below the angle field.

Degrees, radians and the pi key

Radians replace the 360 degree turn with 2 pi, so a quarter turn is pi/2 and one radian is about 57.2958 degrees. The secant of 1 radian is 1.8508, a value with no simple fraction, which is why radian answers are usually written to four decimals. The secant of 7pi/6 is cleaner: 7pi/6 is 210 degrees, the cosine is -0.8660, and the secant is -1.1547.

Mixing the two modes is the single most common error with this function. A calculator left in radian mode will read an input of 60 as 60 radians, which is 60 minus 9 full turns, or 343.94 degrees, and return 1.0507 instead of 2. If a secant answer is close to 1 when you expected a large number, check the mode before checking the formula.

Two more conventions are worth knowing. Secant is even, so sec(-40) equals sec(40), which is useful when a bearing comes in negative. And secant repeats every full turn, so sec(400) equals sec(40). Both follow from cosine, which has the same two properties.

Secant the function is not a secant line

The word shows up twice in math classes with two unrelated meanings. The secant function on this page is a ratio, 1 over cosine, and it takes one angle. A secant line is a straight line through two points on a curve, and its slope is the average rate of change between those points. If you need the second one, the slope calculator computes rise over run from two coordinate pairs, and that is the right tool.

The confusion is old rather than new: both meanings trace back to a line cutting a circle, one as the ray that meets the tangent, the other as a chord between two points. The arithmetic lives in different places, though. A secant line answer has units of y per x, while a secant function answer is a pure number with no units at all.

Related angles have their own pages. The reference angle calculator reduces an angle to its 0 to 90 degree form, the coterminal angle calculator handles full turns, and the right triangle calculator solves the sides this page reads. Use them together when a homework problem asks for the secant of a large angle: coterminal first, reference angle second, secant last.