Arcsine Calculator
Enter a sine value to quickly calculate the corresponding angle in degrees and radians.
Calculate arcsin(x)
Degrees
Radians
What is the Arcsine Function?
The arcsine function, also known as the inverse sine function, is the reverse of the sine function. It is denoted as \(\arcsin(x)\) or \(\sin^{-1}(x)\) and is used to calculate the angle that corresponds to a given sine value. For the sine function \(y = \sin(\theta)\), the arcsine function is defined as: \( \theta = \arcsin(x) \) Here: \(-1 \leq x \leq 1\), \(-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}\). The range of the arcsine function is \([- \frac{\pi}{2}, \frac{\pi}{2}]\), ensuring that the function is unique and invertible.
Examples
Example 1: Find the angle for \(\sin(\theta) = 0.5\)
Solution:
\( \theta = \arcsin(0.5) = \frac{\pi}{6} \approx 0.5236 \, \text{radians} \)
The angle corresponding to a sine value of 0.5 is \(\frac{\pi}{6}\) or 30°.
Example 2: Find the angle for \(\sin(\theta) = 1\)
Solution:
\( \theta = \arcsin(1) = \frac{\pi}{2} \approx 1.5708 \, \text{radians} \)
The angle corresponding to a sine value of 1 is \(\frac{\pi}{2}\) or 90°.
Graph of the Arcsine Function
The graph of the arcsine function is a smooth, monotonic curve that connects points within its defined domain and range. Below are its key features:
- Domain: \([-1, 1]\)
- Range: \([- \frac{\pi}{2}, \frac{\pi}{2}]\)
- Monotonicity: The arcsine function is strictly increasing throughout its domain.
- Odd Function: \(\arcsin(-x) = -\arcsin(x)\), indicating symmetry about the origin.
Arcsine Conversion Table
| Sine Value | Degrees | Radians |
|---|---|---|
| -1 | -90° | \(\frac{-\pi}{2}\) |
| -0.9998477 | -89° | \(\frac{-89\pi}{180}\) |
| -0.99939083 | -88° | \(\frac{-22\pi}{45}\) |
| -0.99862953 | -87° | \(\frac{-29\pi}{60}\) |
| -0.99756405 | -86° | \(\frac{-43\pi}{90}\) |
| -0.9961947 | -85° | \(\frac{-17\pi}{36}\) |
| -0.9945219 | -84° | \(\frac{-7\pi}{15}\) |
| -0.99254615 | -83° | \(\frac{-83\pi}{180}\) |
| -0.99026807 | -82° | \(\frac{-41\pi}{90}\) |
| -0.98768834 | -81° | \(\frac{-9\pi}{20}\) |
| -0.98480775 | -80° | \(\frac{-4\pi}{9}\) |
| -0.98162718 | -79° | \(\frac{-79\pi}{180}\) |
| -0.9781476 | -78° | \(\frac{-13\pi}{30}\) |
| -0.97437006 | -77° | \(\frac{-77\pi}{180}\) |
| -0.97029573 | -76° | \(\frac{-19\pi}{45}\) |
| -0.96592583 | -75° | \(\frac{-5\pi}{12}\) |
| -0.9612617 | -74° | \(\frac{-37\pi}{90}\) |
| -0.95630476 | -73° | \(\frac{-73\pi}{180}\) |
| -0.95105652 | -72° | \(\frac{-2\pi}{5}\) |
| -0.94551858 | -71° | \(\frac{-71\pi}{180}\) |
| -0.93969262 | -70° | \(\frac{-7\pi}{18}\) |
| -0.93358043 | -69° | \(\frac{-23\pi}{60}\) |
| -0.92718385 | -68° | \(\frac{-17\pi}{45}\) |
| -0.92050485 | -67° | \(\frac{-67\pi}{180}\) |
| -0.91354546 | -66° | \(\frac{-11\pi}{30}\) |
| -0.90630779 | -65° | \(\frac{-13\pi}{36}\) |
| -0.89879405 | -64° | \(\frac{-16\pi}{45}\) |
| -0.89100652 | -63° | \(\frac{-7\pi}{20}\) |
| -0.88294759 | -62° | \(\frac{-31\pi}{90}\) |
| -0.87461971 | -61° | \(\frac{-61\pi}{180}\) |
| -0.8660254 | -60° | \(\frac{-\pi}{3}\) |
| -0.8571673 | -59° | \(\frac{-59\pi}{180}\) |
| -0.8480481 | -58° | \(\frac{-29\pi}{90}\) |
| -0.83867057 | -57° | \(\frac{-19\pi}{60}\) |
| -0.82903757 | -56° | \(\frac{-14\pi}{45}\) |
| -0.81915204 | -55° | \(\frac{-11\pi}{36}\) |
| -0.80901699 | -54° | \(\frac{-3\pi}{10}\) |
| -0.79863551 | -53° | \(\frac{-53\pi}{180}\) |
| -0.78801075 | -52° | \(\frac{-13\pi}{45}\) |
| -0.77714596 | -51° | \(\frac{-17\pi}{60}\) |
| -0.76604444 | -50° | \(\frac{-5\pi}{18}\) |
| -0.75470958 | -49° | \(\frac{-49\pi}{180}\) |
| -0.74314483 | -48° | \(\frac{-4\pi}{15}\) |
| -0.7313537 | -47° | \(\frac{-47\pi}{180}\) |
| -0.7193398 | -46° | \(\frac{-23\pi}{90}\) |
| -0.70710678 | -45° | \(\frac{-\pi}{4}\) |
| -0.69465837 | -44° | \(\frac{-11\pi}{45}\) |
| -0.68199836 | -43° | \(\frac{-43\pi}{180}\) |
| -0.66913061 | -42° | \(\frac{-7\pi}{30}\) |
| -0.65605903 | -41° | \(\frac{-41\pi}{180}\) |
| -0.64278761 | -40° | \(\frac{-2\pi}{9}\) |
| -0.62932039 | -39° | \(\frac{-13\pi}{60}\) |
| -0.61566148 | -38° | \(\frac{-19\pi}{90}\) |
| -0.60181502 | -37° | \(\frac{-37\pi}{180}\) |
| -0.58778525 | -36° | \(\frac{-\pi}{5}\) |
| -0.57357644 | -35° | \(\frac{-7\pi}{36}\) |
| -0.5591929 | -34° | \(\frac{-17\pi}{90}\) |
| -0.54463904 | -33° | \(\frac{-11\pi}{60}\) |
| -0.52991926 | -32° | \(\frac{-8\pi}{45}\) |
| -0.51503807 | -31° | \(\frac{-31\pi}{180}\) |
| -0.5 | -30° | \(\frac{-\pi}{6}\) |
| -0.48480962 | -29° | \(\frac{-29\pi}{180}\) |
| -0.46947156 | -28° | \(\frac{-7\pi}{45}\) |
| -0.4539905 | -27° | \(\frac{-3\pi}{20}\) |
| -0.43837115 | -26° | \(\frac{-13\pi}{90}\) |
| -0.42261826 | -25° | \(\frac{-5\pi}{36}\) |
| -0.40673664 | -24° | \(\frac{-2\pi}{15}\) |
| -0.39073113 | -23° | \(\frac{-23\pi}{180}\) |
| -0.37460659 | -22° | \(\frac{-11\pi}{90}\) |
| -0.35836795 | -21° | \(\frac{-7\pi}{60}\) |
| -0.34202014 | -20° | \(\frac{-\pi}{9}\) |
| -0.32556815 | -19° | \(\frac{-19\pi}{180}\) |
| -0.30901699 | -18° | \(\frac{-\pi}{10}\) |
| -0.2923717 | -17° | \(\frac{-17\pi}{180}\) |
| -0.27563736 | -16° | \(\frac{-4\pi}{45}\) |
| -0.25881905 | -15° | \(\frac{-\pi}{12}\) |
| -0.2419219 | -14° | \(\frac{-7\pi}{90}\) |
| -0.22495105 | -13° | \(\frac{-13\pi}{180}\) |
| -0.20791169 | -12° | \(\frac{-\pi}{15}\) |
| -0.190809 | -11° | \(\frac{-11\pi}{180}\) |
| -0.17364818 | -10° | \(\frac{-\pi}{18}\) |
| -0.15643447 | -9° | \(\frac{-\pi}{20}\) |
| -0.1391731 | -8° | \(\frac{-2\pi}{45}\) |
| -0.12186934 | -7° | \(\frac{-7\pi}{180}\) |
| -0.10452846 | -6° | \(\frac{-\pi}{30}\) |
| -0.08715574 | -5° | \(\frac{-\pi}{36}\) |
| -0.06975647 | -4° | \(\frac{-\pi}{45}\) |
| -0.05233596 | -3° | \(\frac{-\pi}{60}\) |
| -0.0348995 | -2° | \(\frac{-\pi}{90}\) |
| -0.01745241 | -1° | \(\frac{-\pi}{180}\) |
| 0 | 0° | 0 |
| 0.01745241 | 1° | \(\frac{\pi}{180}\) |
| 0.0348995 | 2° | \(\frac{\pi}{90}\) |
| 0.05233596 | 3° | \(\frac{\pi}{60}\) |
| 0.06975647 | 4° | \(\frac{\pi}{45}\) |
| 0.08715574 | 5° | \(\frac{\pi}{36}\) |
| 0.10452846 | 6° | \(\frac{\pi}{30}\) |
| 0.12186934 | 7° | \(\frac{7\pi}{180}\) |
| 0.1391731 | 8° | \(\frac{2\pi}{45}\) |
| 0.15643447 | 9° | \(\frac{\pi}{20}\) |
| 0.17364818 | 10° | \(\frac{\pi}{18}\) |
| 0.190809 | 11° | \(\frac{11\pi}{180}\) |
| 0.20791169 | 12° | \(\frac{\pi}{15}\) |
| 0.22495105 | 13° | \(\frac{13\pi}{180}\) |
| 0.2419219 | 14° | \(\frac{7\pi}{90}\) |
| 0.25881905 | 15° | \(\frac{\pi}{12}\) |
| 0.27563736 | 16° | \(\frac{4\pi}{45}\) |
| 0.2923717 | 17° | \(\frac{17\pi}{180}\) |
| 0.30901699 | 18° | \(\frac{\pi}{10}\) |
| 0.32556815 | 19° | \(\frac{19\pi}{180}\) |
| 0.34202014 | 20° | \(\frac{\pi}{9}\) |
| 0.35836795 | 21° | \(\frac{7\pi}{60}\) |
| 0.37460659 | 22° | \(\frac{11\pi}{90}\) |
| 0.39073113 | 23° | \(\frac{23\pi}{180}\) |
| 0.40673664 | 24° | \(\frac{2\pi}{15}\) |
| 0.42261826 | 25° | \(\frac{5\pi}{36}\) |
| 0.43837115 | 26° | \(\frac{13\pi}{90}\) |
| 0.4539905 | 27° | \(\frac{3\pi}{20}\) |
| 0.46947156 | 28° | \(\frac{7\pi}{45}\) |
| 0.48480962 | 29° | \(\frac{29\pi}{180}\) |
| 0.5 | 30° | \(\frac{\pi}{6}\) |
| 0.51503807 | 31° | \(\frac{31\pi}{180}\) |
| 0.52991926 | 32° | \(\frac{8\pi}{45}\) |
| 0.54463904 | 33° | \(\frac{11\pi}{60}\) |
| 0.5591929 | 34° | \(\frac{17\pi}{90}\) |
| 0.57357644 | 35° | \(\frac{7\pi}{36}\) |
| 0.58778525 | 36° | \(\frac{\pi}{5}\) |
| 0.60181502 | 37° | \(\frac{37\pi}{180}\) |
| 0.61566148 | 38° | \(\frac{19\pi}{90}\) |
| 0.62932039 | 39° | \(\frac{13\pi}{60}\) |
| 0.64278761 | 40° | \(\frac{2\pi}{9}\) |
| 0.65605903 | 41° | \(\frac{41\pi}{180}\) |
| 0.66913061 | 42° | \(\frac{7\pi}{30}\) |
| 0.68199836 | 43° | \(\frac{43\pi}{180}\) |
| 0.69465837 | 44° | \(\frac{11\pi}{45}\) |
| 0.70710678 | 45° | \(\frac{\pi}{4}\) |
| 0.7193398 | 46° | \(\frac{23\pi}{90}\) |
| 0.7313537 | 47° | \(\frac{47\pi}{180}\) |
| 0.74314483 | 48° | \(\frac{4\pi}{15}\) |
| 0.75470958 | 49° | \(\frac{49\pi}{180}\) |
| 0.76604444 | 50° | \(\frac{5\pi}{18}\) |
| 0.77714596 | 51° | \(\frac{17\pi}{60}\) |
| 0.78801075 | 52° | \(\frac{13\pi}{45}\) |
| 0.79863551 | 53° | \(\frac{53\pi}{180}\) |
| 0.80901699 | 54° | \(\frac{3\pi}{10}\) |
| 0.81915204 | 55° | \(\frac{11\pi}{36}\) |
| 0.82903757 | 56° | \(\frac{14\pi}{45}\) |
| 0.83867057 | 57° | \(\frac{19\pi}{60}\) |
| 0.8480481 | 58° | \(\frac{29\pi}{90}\) |
| 0.8571673 | 59° | \(\frac{59\pi}{180}\) |
| 0.8660254 | 60° | \(\frac{\pi}{3}\) |
| 0.87461971 | 61° | \(\frac{61\pi}{180}\) |
| 0.88294759 | 62° | \(\frac{31\pi}{90}\) |
| 0.89100652 | 63° | \(\frac{7\pi}{20}\) |
| 0.89879405 | 64° | \(\frac{16\pi}{45}\) |
| 0.90630779 | 65° | \(\frac{13\pi}{36}\) |
| 0.91354546 | 66° | \(\frac{11\pi}{30}\) |
| 0.92050485 | 67° | \(\frac{67\pi}{180}\) |
| 0.92718385 | 68° | \(\frac{17\pi}{45}\) |
| 0.93358043 | 69° | \(\frac{23\pi}{60}\) |
| 0.93969262 | 70° | \(\frac{7\pi}{18}\) |
| 0.94551858 | 71° | \(\frac{71\pi}{180}\) |
| 0.95105652 | 72° | \(\frac{2\pi}{5}\) |
| 0.95630476 | 73° | \(\frac{73\pi}{180}\) |
| 0.9612617 | 74° | \(\frac{37\pi}{90}\) |
| 0.96592583 | 75° | \(\frac{5\pi}{12}\) |
| 0.97029573 | 76° | \(\frac{19\pi}{45}\) |
| 0.97437006 | 77° | \(\frac{77\pi}{180}\) |
| 0.9781476 | 78° | \(\frac{13\pi}{30}\) |
| 0.98162718 | 79° | \(\frac{79\pi}{180}\) |
| 0.98480775 | 80° | \(\frac{4\pi}{9}\) |
| 0.98768834 | 81° | \(\frac{9\pi}{20}\) |
| 0.99026807 | 82° | \(\frac{41\pi}{90}\) |
| 0.99254615 | 83° | \(\frac{83\pi}{180}\) |
| 0.9945219 | 84° | \(\frac{7\pi}{15}\) |
| 0.9961947 | 85° | \(\frac{17\pi}{36}\) |
| 0.99756405 | 86° | \(\frac{43\pi}{90}\) |
| 0.99862953 | 87° | \(\frac{29\pi}{60}\) |
| 0.99939083 | 88° | \(\frac{22\pi}{45}\) |
| 0.9998477 | 89° | \(\frac{89\pi}{180}\) |
| 1 | 90° | \(\frac{\pi}{2}\) |