Inverse Trig Calculator
With ratio 0.5, inverse trig comes to 30.0000 ° — angle. It is reached in 3 steps, the last of which is 0.5235988 * 180 / pi, and each one is printed on the page with its numbers filled in. The formula is the one published by NIST Digital Library of Mathematical Functions §4.23, not an approximation fitted to it.
Arcsine, arccosine and arctangent: the angle whose sine, cosine or tangent is the ratio you have, in degrees and radians.
Formula and sources checked · How we check
Find the angle whose Sine is (arcsin), Ratio 0.5
30.0000 °
Angle for the example below. Editing a field recomputes the calculator below; this figure holds the answer the page was loaded with.
It is written into the HTML rather than drawn by a script, so a search engine reading this page without running JavaScript still finds an answer.
- Angle in radians
(asin(0.5))0.524- Angle in degrees
0.5235988 * 180 / pi30 °- Complement (90° − angle)
90 - 3060 °
Worked example
Arcsine of 0.5 is 30 degrees, or π/6 radians. So is arcsine of 0.5 at 150 degrees — the sine repeats — but the inverse function has to return one answer, and by convention it returns the one between −90 and 90.
How to work it out yourself
- 1.Feed it a ratio, not an angle. Opposite over hypotenuse goes into arcsine; adjacent over hypotenuse into arccosine; opposite over adjacent into arctangent.
- 2.A ratio outside −1 to 1 has no arcsine or arccosine. If a triangle produced one, a side was measured wrong or the wrong pair was divided.
- 3.Only arctangent accepts any value, because the tangent itself is unbounded. A slope of 12 in 12 is arctan(1) = 45 degrees.
The formula
- Angle in radians
(asin(0.5)) - Angle in degrees
0.5235988 * 180 / pi - Complement (90° − angle)
90 - 30
Source: NIST Digital Library of Mathematical Functions §4.23 — inverse trigonometric functions, NIST DLMF §4.14 — trigonometric functions
Questions people actually ask
- Why does arcsine only return angles between −90 and 90?
- Because the sine takes every value twice per turn, so the inverse has to pick a branch to be a function at all. The chosen range is −90 to 90 for arcsine and arctangent, and 0 to 180 for arccosine. If the angle you want is in another quadrant, the sign of the other two sides is what tells you which.
- What is the difference between arcsin and sin⁻¹?
- Nothing — they are two notations for the same function. The superscript is not a reciprocal: 1/sin x is the cosecant, which is a different thing entirely.
- How do I get an angle from a slope or a grade?
- A grade is rise over run, which is a tangent, so the angle is arctan(grade). A 6% grade is arctan(0.06) = 3.4336 degrees. Note that a road sign quotes the percentage, not the angle.
Related
- Sin Cos Tan CalculatorSine, cosine and tangent of an angle in degrees or radians, with the angle also shown in the other unit and the reciprocal functions.
- Slope & Grade CalculatorGrade percentage, angle in degrees, ratio and roof pitch from a rise and a run.
- Law of Cosines CalculatorSolves a triangle from two sides and the angle between them: the third side, the other two angles and the area, using c² = a² + b² − 2ab·cos C.
- Diagonal & Rafter Length CalculatorHypotenuse of a right triangle in feet and inches — brace lengths, rafter runs, and squaring a layout.
- Circle CalculatorArea, circumference and radius of a circle from its diameter — in square feet and square inches.
- Tank & Cylinder Volume CalculatorVolume of a cylinder in cubic feet, US gallons and liters — round tanks, pools, culverts and post holes.
Put this calculator on your site
Free, no attribution required beyond the link.
<iframe src="https://rulecalculators.com/embed/inverse-trig" width="100%" height="420" style="border:1px solid #e7e4de;border-radius:12px" title="Inverse Trig Calculator"></iframe>