Secant (sec) is the trigonometric function equal to 1 divided by cosine: sec θ = 1cos θ. In a right triangle it is the hypotenuse divided by the side adjacent to the angle. Secant is undefined wherever cosine is 0, such as at 90°, and its values are always at least 1 or at most −1.
In Spanish
secante
Also called
sec, secant function, secant ratio
How to say it
SEE-kant
Grade level
Precalculus
Algebra 2PrecalculusCalculusACTAPACCUPLACER
Secant in a picture
Where cosine is small, secant is large; where cosine is 0, secant is undefined. The two graphs touch where cosine is 1 or −1.
Formal definition and notation
For any angle θ with cos θ ≠ 0, sec θ = 1cos θ. For an acute angle of a right triangle this is hypotenuseadjacent; on the unit circle, where the point at angle θ is (x, y), it is 1x. The function y = secx has domain all real numbers except π2 + kπ (vertical asymptotes there), range y ≤ −1 or y ≥ 1, and period2π. It satisfies the identity 1 + tan2θ = sec2θ.
Secant goes with cosine; cosecant goes with sine. Each reciprocal pair has exactly one “co”.
✗ MistakeUsing the inverse key: sec 0.5 is found with cos−1(0.5) = 60°.
✓ Insteadsecx = 1cosx, a reciprocal. cos−1 gives an angle, not a secant.
The −1 on a function name means inverse function, not reciprocal.
✗ MistakeSaying sec 90° = 0.
✓ Insteadcos 90° = 0, so sec 90° = 10 is undefined.
Dividing by 0 is undefined; the graph has a vertical asymptote there.
Not to be confused with
Secantvs.Secant line
In geometry, a secant is a line that cuts a circle at two points. The trig secant is a function, 1cos θ. The names are related historically but mean different things on a test.
Secantvs.Cosecant
Cosecant is 1sin θ, hypotenuse over opposite. Secant is 1cos θ, hypotenuse over adjacent.
Secantvs.Inverse cosine
cos−1(0.5) = 60° is an angle. sec 60° = 2 is a ratio.
Quick check: 3 questions
Question 1 of 3
Which expression equals sec θ?
Show answer and explanation
Answer: B. Secant is the reciprocal of cosine. 1sin θ is cosecant, cos−1θ is inverse cosine, and sin θcos θ is tangent.
Question 2 of 3
What is sec 0°?
Show answer and explanation
Answer: D.cos 0° = 1, so sec 0° = 11 = 1. It is sec 90° that is undefined, and −1 is sec 180°.
Question 3 of 3
Which number can never be a value of sec θ?
Show answer and explanation
Answer: A. Since |cos θ| ≤ 1, its reciprocal satisfies |sec θ| ≥ 1, so 0.5 is impossible. sec 180° = −1, sec 60° = 2, and −3 occurs in Quadrant II or III.
Secant: frequently asked questions
Is secant the same as 1 over cosine?
Yes. By definition sec θ = 1cos θ, as long as cos θ ≠ 0. That is why most calculators have no sec key, and you work it out in two steps: find the cosine, then take its reciprocal. For example, cos 45° ≈ 0.7071, so sec 45° ≈ 1.414, which is √2.
Where is secant undefined?
Wherever cosine equals 0: at 90°, 270° and every angle that differs from these by a multiple of 180°, which is π2 + kπ in radians. At those angles the graph of y = secx has vertical asymptotes, and the values shoot up toward infinity or down toward negative infinity.
Acute AngleAn acute angle is an angle that measures more than 0 degrees and less than 90 degrees. It is narrower than the square corner of a page. Angles of 15, 45 and 89 degrees are all acute; 90 degrees is a right angle, so it is not acute. Every triangle has at least two acute angles.Geometry · Grade 4, Grade 5, Grade 7Right TriangleA right triangle is a triangle with one right angle, an angle of exactly 90 degrees. The two sides that form the right angle are the legs, and the longest side, across from the right angle, is the hypotenuse. Its sides follow the Pythagorean theorem: a2 + b2 = c2.Geometry · Grade 4, Grade 6, Grade 8Unit CircleThe unit circle is the circle with radius 1 centered at the origin, x2 + y2 = 1. An angle θ measured from the positive x-axis meets the circle at the point (cos θ, sin θ). This lets sine and cosine work for every angle, including obtuse and negative angles, not just the acute angles of right triangles.Trigonometry · Algebra 2, PrecalculusDomainThe domain of a function is the set of all input values (usually x-values) the function can accept. Unless a problem says otherwise, it is every real number except values that would divide by zero or take an even root of a negative number. On a graph, the domain is how far the graph reaches left and right.Functions · Grade 8, Algebra 1, Algebra 2Real NumberA real number is any number that can be placed on the number line. The real numbers are all the rational numbers (fractions, integers, terminating and repeating decimals) together with all the irrational numbers (such as √2 and π). Square roots of negative numbers, like √−4, are not real.Number System · Grade 8, Algebra 1AsymptoteAn asymptote is a line that a graph gets closer and closer to as it heads off toward infinity. A vertical asymptote x = a appears where the function blows up, often at a zero denominator. A horizontal asymptote y = b shows the value it approaches far left or right. y = 1x has both: x = 0 and y = 0.Functions · Algebra 2, Precalculus, CalculusPeriodThe period of a repeating function is the horizontal length of one complete cycle: the smallest distance after which the graph repeats exactly. y = sinx and y = cosx have period 2π, or 360°, and y = tanx has period π. For y = sin(Bx) or y = cos(Bx), the period is 2π|B|.Trigonometry · Algebra 2, PrecalculusUndefinedUndefined describes an expression that has no value, most often because it divides by zero. 70 is undefined, since no number times 0 gives 7. A vertical line has an undefined slope, and 3x − 2 is undefined at x = 2. Undefined does not mean zero: 07 = 0, but 70 has no value.Problem Solving · Grade 6, Grade 8, Algebra 1HypotenuseThe hypotenuse is the longest side of a right triangle. It is always across from the right angle, and the two shorter sides that form the right angle are the legs. In the Pythagorean theorem a2 + b2 = c2, the hypotenuse is c: in a 3-4-5 right triangle, it is the side of length 5.Geometry · Grade 8, Algebra 1SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2CosineCosine is the trigonometric ratio that compares the side adjacent to an angle with the hypotenuse of a right triangle: cos θ = adjacenthypotenuse. On the unit circle, cos θ is the x-coordinate of the point at angle θ. The cosine of an angle equals the sine of its complement.Trigonometry · Geometry, Algebra 2, PrecalculusQuadrantA quadrant is one of the four regions that the x-axis and y-axis split the coordinate plane into. They are numbered I, II, III and IV counterclockwise, starting at the top right. The signs of a point’s coordinates tell its quadrant: (+, +) is I, (−, +) is II, (−, −) is III and (+, −) is IV.Geometry · Grade 5, Grade 6, Grade 7SOH CAH TOASOH CAH TOA is a memory aid for the three basic trigonometric ratios in a right triangle: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, and Tangent is Opposite over Adjacent. Name the sides from the angle you are using, choose the ratio that links the sides you know and need, then solve.Trigonometry · Geometry, Algebra 2, PrecalculusSineSine is the trigonometric ratio that compares the side opposite an angle with the hypotenuse of a right triangle: sin θ = oppositehypotenuse. On the unit circle, sin θ is the y-coordinate of the point at angle θ, so sine values always stay between −1 and 1.Trigonometry · Geometry, Algebra 2, PrecalculusTangentTangent (tan) is the trigonometric ratio of the side opposite an acute angle to the side adjacent to it in a right triangle: tan θ = oppositeadjacent. For legs 3 and 4, tan θ = 34 = 0.75. It is the TOA in SOH CAH TOA, and it equals sin θcos θ.Trigonometry · Geometry, Algebra 2, PrecalculusTrigonometryTrigonometry is the branch of math that studies how the angles and the side lengths of triangles are related. Its three main ratios, sine, cosine and tangent, compare two sides of a right triangle. Knowing one side and one acute angle, or two sides, you can find every other side and angle without measuring them.Trigonometry · Geometry, Algebra 2, Precalculus