Secant Calculator – Calculate sec(x)
Find the secant of an angle using the sec calculator below. Start by entering the angle in degrees or radians.
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How to Find the Secant of an Angle
In a right triangle, the secant of angle α, or sec(α), is the ratio between the angle’s adjacent side and the hypotenuse.
Secant is a trigonometric function abbreviated sec. Use the formula below to calculate the secant of an angle.
Secant Formula
The secant formula is:
sec(α) = hypotenuse c / adjacent b
Thus, the secant of angle α in a right triangle is equal to the length of the hypotenuse c divided by the adjacent side b.
To solve sec, simply enter the length of the hypotenuse and adjacent side, then solve.
This formula might look very similar to the formula to calculate cosine. That’s because secant is the reciprocal of cosine.
Secant should not be confused with arccos, which is the inverse of the cosine function. The difference being that secant is equal to 1 / cos(x), while arccos is the inverse of the cosine function.
sec(x) = 1 / cos(x) = cos(x)-1
arccos(y) x where y = cos(x)
For example, let’s calculate the secant of angle α in a triangle with the length of the hypotenuse equal to 8 and the adjacent side equal to 6.
sec(α) = 8 / 6
sec(α) = 4 / 3
Secant Graph
If you graph the secant function for every possible angle, it forms a series of repeating U-curves.
One important property to note in the graph above is that the secant of an angle is never in the range of -1 to 1; it’s always smaller than or equal to -1 or larger than or equal to 1.
You’ll also notice that the curves never cross the x-axis at an even multiple of 1/2π + π radians, or 90° + 180°
Secant Table
The table below shows common angles and the sec value for each of them.
| Angle (degrees) | Angle (radians) | Secant |
|---|---|---|
| 0° | 0 | 1 |
| 15° | π / 12 | √6 – √2 |
| 30° | π / 6 | 2√3 / 3 |
| 45° | π / 4 | √2 |
| 60° | π / 3 | 2 |
| 75° | 5π / 12 | √6 + √2 |
| 90° | π / 2 | undefined |
| 105° | 7π / 12 | -√6 – √2 |
| 120° | 2π / 3 | -2 |
| 135° | 3π / 4 | -√2 |
| 150° | 5π / 6 | –2√3 / 3 |
| 165° | 11π / 12 | -√6 – √2 |
| 180° | π | -1 |
| 195° | 13π / 12 | -√6 – √2 |
| 210° | 7π / 6 | –2√3 / 3 |
| 225° | 5π / 4 | -√2 |
| 240° | 4π / 3 | -2 |
| 255° | 17π / 12 | -√6 – √2 |
| 270° | 3π / 2 | undefined |
| 285° | 19π / 12 | √6 + √2 |
| 300° | 5π / 3 | 2 |
| 315° | 7π / 4 | √2 |
| 330° | 11π / 6 | 2√3 / 3 |
| 345° | 23π / 12 | √6 – √2 |
| 360° | 2π | undefined |
You might also be interested in our cosecant and cotangent calculators.