Arcsecant Calculator
Input a secant value to compute the corresponding angle in degrees and radians.
Calculate arcsec(x)
Degrees
Radians
What is the Arcsecant Function?
The arcsecant function, also called the inverse secant function, is the reverse of the secant function. It is typically represented by \(\operatorname{arcsec}(x)\) or \(\sec^{-1}(x)\). This function is used to calculate the angle corresponding to a given secant value. For the secant function \(y = \sec(\theta)\), the arcsecant function is defined as: \( \theta = \operatorname{arcsec}(x) \) Where: \(x \leq -1\) or \(x \geq 1\) (secant values outside of \(-1, 1\)), \(\theta\) lies within the range \([0, \pi]\) but excludes \(\frac{\pi}{2}\). This definition ensures that the arcsecant function is unique and invertible.
Examples
Example 1: Find the angle for \(\sec(\theta) = 2\)
Solution:
\( \theta = \operatorname{arcsec}(2) \approx 1.047 \, \text{radians} \)
The angle corresponding to a secant value of 2 is approximately 1.047 radians or 60°.
Example 2: Find the angle for \(\sec(\theta) = -2\)
Solution:
\( \theta = \operatorname{arcsec}(-2) \approx 2.094 \, \text{radians} \)
The angle corresponding to a secant value of -2 is approximately 2.094 radians or 120°.
Graph of the Arcsecant Function
The graph of the arcsecant function consists of two monotonic increasing curve segments: \((-\infty, -1] \cup [1, +\infty)\). Key characteristics of the graph include:
- Domain: \((-\infty, -1] \cup [1, +\infty)\)
- Range: \((0, \pi]\) (excluding \(\frac{\pi}{2}\))
- Monotonicity: The function is strictly increasing in each part of its domain.
- Symmetry: Arcsecant satisfies the property \(\operatorname{arcsec}(-x) = \pi - \operatorname{arcsec}(x)\).
Arcsecant Conversion Table
| Secant Value | Degrees | Radians |
|---|---|---|
| 1 | 0° | 0 |
| 1.00015233 | 1° | \(\frac{\pi}{180}\) |
| 1.00060954 | 2° | \(\frac{\pi}{90}\) |
| 1.00137235 | 3° | \(\frac{\pi}{60}\) |
| 1.0024419 | 4° | \(\frac{\pi}{45}\) |
| 1.00381984 | 5° | \(\frac{\pi}{36}\) |
| 1.00550828 | 6° | \(\frac{\pi}{30}\) |
| 1.00750983 | 7° | \(\frac{7\pi}{180}\) |
| 1.00982757 | 8° | \(\frac{2\pi}{45}\) |
| 1.01246513 | 9° | \(\frac{\pi}{20}\) |
| 1.01542661 | 10° | \(\frac{\pi}{18}\) |
| 1.01871669 | 11° | \(\frac{11\pi}{180}\) |
| 1.02234059 | 12° | \(\frac{\pi}{15}\) |
| 1.02630411 | 13° | \(\frac{13\pi}{180}\) |
| 1.03061363 | 14° | \(\frac{7\pi}{90}\) |
| 1.03527618 | 15° | \(\frac{\pi}{12}\) |
| 1.04029944 | 16° | \(\frac{4\pi}{45}\) |
| 1.04569176 | 17° | \(\frac{17\pi}{180}\) |
| 1.05146222 | 18° | \(\frac{\pi}{10}\) |
| 1.05762068 | 19° | \(\frac{19\pi}{180}\) |
| 1.06417777 | 20° | \(\frac{\pi}{9}\) |
| 1.07114499 | 21° | \(\frac{7\pi}{60}\) |
| 1.07853474 | 22° | \(\frac{11\pi}{90}\) |
| 1.08636038 | 23° | \(\frac{23\pi}{180}\) |
| 1.09463628 | 24° | \(\frac{2\pi}{15}\) |
| 1.10337792 | 25° | \(\frac{5\pi}{36}\) |
| 1.11260194 | 26° | \(\frac{13\pi}{90}\) |
| 1.12232624 | 27° | \(\frac{3\pi}{20}\) |
| 1.13257005 | 28° | \(\frac{7\pi}{45}\) |
| 1.14335407 | 29° | \(\frac{29\pi}{180}\) |
| 1.15470054 | 30° | \(\frac{\pi}{6}\) |
| 1.1666334 | 31° | \(\frac{31\pi}{180}\) |
| 1.1791784 | 32° | \(\frac{8\pi}{45}\) |
| 1.19236329 | 33° | \(\frac{11\pi}{60}\) |
| 1.20621795 | 34° | \(\frac{17\pi}{90}\) |
| 1.22077459 | 35° | \(\frac{7\pi}{36}\) |
| 1.23606798 | 36° | \(\frac{\pi}{5}\) |
| 1.25213566 | 37° | \(\frac{37\pi}{180}\) |
| 1.26901822 | 38° | \(\frac{19\pi}{90}\) |
| 1.28675957 | 39° | \(\frac{13\pi}{60}\) |
| 1.30540729 | 40° | \(\frac{2\pi}{9}\) |
| 1.32501299 | 41° | \(\frac{41\pi}{180}\) |
| 1.34563273 | 42° | \(\frac{7\pi}{30}\) |
| 1.36732746 | 43° | \(\frac{43\pi}{180}\) |
| 1.39016359 | 44° | \(\frac{11\pi}{45}\) |
| 1.41421356 | 45° | \(\frac{\pi}{4}\) |
| 1.43955654 | 46° | \(\frac{23\pi}{90}\) |
| 1.46627919 | 47° | \(\frac{47\pi}{180}\) |
| 1.49447655 | 48° | \(\frac{4\pi}{15}\) |
| 1.52425309 | 49° | \(\frac{49\pi}{180}\) |
| 1.55572383 | 50° | \(\frac{5\pi}{18}\) |
| 1.58901573 | 51° | \(\frac{17\pi}{60}\) |
| 1.62426925 | 52° | \(\frac{13\pi}{45}\) |
| 1.66164014 | 53° | \(\frac{53\pi}{180}\) |
| 1.70130162 | 54° | \(\frac{3\pi}{10}\) |
| 1.7434468 | 55° | \(\frac{11\pi}{36}\) |
| 1.78829165 | 56° | \(\frac{14\pi}{45}\) |
| 1.83607846 | 57° | \(\frac{19\pi}{60}\) |
| 1.88707991 | 58° | \(\frac{29\pi}{90}\) |
| 1.94160403 | 59° | \(\frac{59\pi}{180}\) |
| 2 | 60° | \(\frac{\pi}{3}\) |
| 2.06266534 | 61° | \(\frac{61\pi}{180}\) |
| 2.13005447 | 62° | \(\frac{31\pi}{90}\) |
| 2.20268926 | 63° | \(\frac{7\pi}{20}\) |
| 2.28117203 | 64° | \(\frac{16\pi}{45}\) |
| 2.36620158 | 65° | \(\frac{13\pi}{36}\) |
| 2.45859334 | 66° | \(\frac{11\pi}{30}\) |
| 2.55930467 | 67° | \(\frac{67\pi}{180}\) |
| 2.66946716 | 68° | \(\frac{17\pi}{45}\) |
| 2.79042811 | 69° | \(\frac{23\pi}{60}\) |
| 2.9238044 | 70° | \(\frac{7\pi}{18}\) |
| 3.07155349 | 71° | \(\frac{71\pi}{180}\) |
| 3.23606798 | 72° | \(\frac{2\pi}{5}\) |
| 3.42030362 | 73° | \(\frac{73\pi}{180}\) |
| 3.62795528 | 74° | \(\frac{37\pi}{90}\) |
| 3.86370331 | 75° | \(\frac{5\pi}{12}\) |
| 4.13356549 | 76° | \(\frac{19\pi}{45}\) |
| 4.44541148 | 77° | \(\frac{77\pi}{180}\) |
| 4.80973434 | 78° | \(\frac{13\pi}{30}\) |
| 5.24084306 | 79° | \(\frac{79\pi}{180}\) |
| 5.75877048 | 80° | \(\frac{4\pi}{9}\) |
| 6.39245322 | 81° | \(\frac{9\pi}{20}\) |
| 7.18529653 | 82° | \(\frac{41\pi}{90}\) |
| 8.20550905 | 83° | \(\frac{83\pi}{180}\) |
| 9.56677223 | 84° | \(\frac{7\pi}{15}\) |
| 11.47371325 | 85° | \(\frac{17\pi}{36}\) |
| 14.33558703 | 86° | \(\frac{43\pi}{90}\) |
| 19.10732261 | 87° | \(\frac{29\pi}{60}\) |
| 28.65370835 | 88° | \(\frac{22\pi}{45}\) |
| 57.2986885 | 89° | \(\frac{89\pi}{180}\) |
| -57.2986885 | 91° | \(\frac{91\pi}{180}\) |
| -28.65370835 | 92° | \(\frac{23\pi}{45}\) |
| -19.10732261 | 93° | \(\frac{31\pi}{60}\) |
| -14.33558703 | 94° | \(\frac{47\pi}{90}\) |
| -11.47371325 | 95° | \(\frac{19\pi}{36}\) |
| -9.56677223 | 96° | \(\frac{8\pi}{15}\) |
| -8.20550905 | 97° | \(\frac{97\pi}{180}\) |
| -7.18529653 | 98° | \(\frac{49\pi}{90}\) |
| -6.39245322 | 99° | \(\frac{11\pi}{20}\) |
| -5.75877048 | 100° | \(\frac{5\pi}{9}\) |
| -5.24084306 | 101° | \(\frac{101\pi}{180}\) |
| -4.80973434 | 102° | \(\frac{17\pi}{30}\) |
| -4.44541148 | 103° | \(\frac{103\pi}{180}\) |
| -4.13356549 | 104° | \(\frac{26\pi}{45}\) |
| -3.86370331 | 105° | \(\frac{7\pi}{12}\) |
| -3.62795528 | 106° | \(\frac{53\pi}{90}\) |
| -3.42030362 | 107° | \(\frac{107\pi}{180}\) |
| -3.23606798 | 108° | \(\frac{3\pi}{5}\) |
| -3.07155349 | 109° | \(\frac{109\pi}{180}\) |
| -2.9238044 | 110° | \(\frac{11\pi}{18}\) |
| -2.79042811 | 111° | \(\frac{37\pi}{60}\) |
| -2.66946716 | 112° | \(\frac{28\pi}{45}\) |
| -2.55930467 | 113° | \(\frac{113\pi}{180}\) |
| -2.45859334 | 114° | \(\frac{19\pi}{30}\) |
| -2.36620158 | 115° | \(\frac{23\pi}{36}\) |
| -2.28117203 | 116° | \(\frac{29\pi}{45}\) |
| -2.20268926 | 117° | \(\frac{13\pi}{20}\) |
| -2.13005447 | 118° | \(\frac{59\pi}{90}\) |
| -2.06266534 | 119° | \(\frac{119\pi}{180}\) |
| -2 | 120° | \(\frac{2\pi}{3}\) |
| -1.94160403 | 121° | \(\frac{121\pi}{180}\) |
| -1.88707991 | 122° | \(\frac{61\pi}{90}\) |
| -1.83607846 | 123° | \(\frac{41\pi}{60}\) |
| -1.78829165 | 124° | \(\frac{31\pi}{45}\) |
| -1.7434468 | 125° | \(\frac{25\pi}{36}\) |
| -1.70130162 | 126° | \(\frac{7\pi}{10}\) |
| -1.66164014 | 127° | \(\frac{127\pi}{180}\) |
| -1.62426925 | 128° | \(\frac{32\pi}{45}\) |
| -1.58901573 | 129° | \(\frac{43\pi}{60}\) |
| -1.55572383 | 130° | \(\frac{13\pi}{18}\) |
| -1.52425309 | 131° | \(\frac{131\pi}{180}\) |
| -1.49447655 | 132° | \(\frac{11\pi}{15}\) |
| -1.46627919 | 133° | \(\frac{133\pi}{180}\) |
| -1.43955654 | 134° | \(\frac{67\pi}{90}\) |
| -1.41421356 | 135° | \(\frac{3\pi}{4}\) |
| -1.39016359 | 136° | \(\frac{34\pi}{45}\) |
| -1.36732746 | 137° | \(\frac{137\pi}{180}\) |
| -1.34563273 | 138° | \(\frac{23\pi}{30}\) |
| -1.32501299 | 139° | \(\frac{139\pi}{180}\) |
| -1.30540729 | 140° | \(\frac{7\pi}{9}\) |
| -1.28675957 | 141° | \(\frac{47\pi}{60}\) |
| -1.26901822 | 142° | \(\frac{71\pi}{90}\) |
| -1.25213566 | 143° | \(\frac{143\pi}{180}\) |
| -1.23606798 | 144° | \(\frac{4\pi}{5}\) |
| -1.22077459 | 145° | \(\frac{29\pi}{36}\) |
| -1.20621795 | 146° | \(\frac{73\pi}{90}\) |
| -1.19236329 | 147° | \(\frac{49\pi}{60}\) |
| -1.1791784 | 148° | \(\frac{37\pi}{45}\) |
| -1.1666334 | 149° | \(\frac{149\pi}{180}\) |
| -1.15470054 | 150° | \(\frac{5\pi}{6}\) |
| -1.14335407 | 151° | \(\frac{151\pi}{180}\) |
| -1.13257005 | 152° | \(\frac{38\pi}{45}\) |
| -1.12232624 | 153° | \(\frac{17\pi}{20}\) |
| -1.11260194 | 154° | \(\frac{77\pi}{90}\) |
| -1.10337792 | 155° | \(\frac{31\pi}{36}\) |
| -1.09463628 | 156° | \(\frac{13\pi}{15}\) |
| -1.08636038 | 157° | \(\frac{157\pi}{180}\) |
| -1.07853474 | 158° | \(\frac{79\pi}{90}\) |
| -1.07114499 | 159° | \(\frac{53\pi}{60}\) |
| -1.06417777 | 160° | \(\frac{8\pi}{9}\) |
| -1.05762068 | 161° | \(\frac{161\pi}{180}\) |
| -1.05146222 | 162° | \(\frac{9\pi}{10}\) |
| -1.04569176 | 163° | \(\frac{163\pi}{180}\) |
| -1.04029944 | 164° | \(\frac{41\pi}{45}\) |
| -1.03527618 | 165° | \(\frac{11\pi}{12}\) |
| -1.03061363 | 166° | \(\frac{83\pi}{90}\) |
| -1.02630411 | 167° | \(\frac{167\pi}{180}\) |
| -1.02234059 | 168° | \(\frac{14\pi}{15}\) |
| -1.01871669 | 169° | \(\frac{169\pi}{180}\) |
| -1.01542661 | 170° | \(\frac{17\pi}{18}\) |
| -1.01246513 | 171° | \(\frac{19\pi}{20}\) |
| -1.00982757 | 172° | \(\frac{43\pi}{45}\) |
| -1.00750983 | 173° | \(\frac{173\pi}{180}\) |
| -1.00550828 | 174° | \(\frac{29\pi}{30}\) |
| -1.00381984 | 175° | \(\frac{35\pi}{36}\) |
| -1.0024419 | 176° | \(\frac{44\pi}{45}\) |
| -1.00137235 | 177° | \(\frac{59\pi}{60}\) |
| -1.00060954 | 178° | \(\frac{89\pi}{90}\) |
| -1.00015233 | 179° | \(\frac{179\pi}{180}\) |
| -1 | 180° | π |