Inverse Trig (Ratio to Angle)

This is the calculation for when you know a side ratio and want the angle. For the same 0.6, asin gives 36.87°, acos 53.13° and atan 30.96° — so decide which ratio you have first.

Your numbers

Ratio

Angle (°)

36.87°

cos

53.13°

tan

30.964°

Radians

0.6435

Formula

Angle (°) = asin(Ratio)

asin and acos only accept values from −1 to 1. atan takes any real number but never quite reaches 90°.

What to enter

Input	Default	Accepted range
RatioThe trigonometric ratio to invert, between −1 and 1.	0.6	-1 ~ 1

Step by step

FormulaAngle (°) = asin(Ratio)

With the default numbersAngle (°) = asin(0.6)

AnswerAngle (°) = 36.87 °

Quick reference table

Results when only Ratio changes and everything else stays put.

Ratio	Angle (°) (°)	cos (°)	tan (°)
0.3	17.458	72.542	16.699
0.45	26.744	63.256	24.228
0.6	36.87	53.13	30.964
0.9	64.158	25.842	41.987
1	90	0	45

What each result means

Result	At default values
Angle (°) (°)An angle in degrees; a full turn is 360°.	36.87
cos (°)The cosine of the angle — adjacent over hypotenuse.	53.13
tan (°)The tangent of the angle; it is undefined at 90°.	30.964
RadiansAn angle in radians; a full turn is 2π.	0.6435

Common mistakes

asin and acos only accept values from −1 to 1. atan takes any real number but never quite reaches 90°.

Glossary

Ratio: The trigonometric ratio to invert, between −1 and 1.
Angle (°): An angle in degrees; a full turn is 360°.
cos: The cosine of the angle — adjacent over hypotenuse.
tan: The tangent of the angle; it is undefined at 90°.
Radians: An angle in radians; a full turn is 2π.

Frequently asked questions

QHow is Inverse Trig (Ratio to Angle) calculated?

Angle (°) = asin(Ratio) — This is the calculation for when you know a side ratio and want the angle. For the same 0.6, asin gives 36.87°, acos 53.13° and atan 30.96° — so decide which ratio you have first.

QCan you walk through an example?

With Ratio 0.6, the answer is Angle (°) 36.87°.

QWhat do I need to enter?

Enter Ratio. The result recalculates as you type, and an empty box counts as zero.

QHow much does the answer move if I change a number?

Changing only Ratio moves the answer to Ratio 0.3 → Angle (°) (°) 17.458 and Ratio 1 → Angle (°) (°) 90. The table below lays out five steps.

QHow are the numbers rounded?

Money is shown to the nearest whole unit, percentages to one decimal place and everything else to two. What you see is rounded; the calculation itself carries the unrounded value forward.

QAnything to watch out for?

asin and acos only accept values from −1 to 1. atan takes any real number but never quite reaches 90°.

Related calculators

Pythagorean TheoremFind the hypotenuse from the two legs of a right triangle.Missing Leg of a Right TriangleFind the remaining side from the hypotenuse and one leg.Angle from Two SidesGet the angle of a slope from its rise and run.Trig Function ValuesEnter an angle and see sine, cosine and tangent at once.

Keep your length units consistent across inputs — mixing centimetres and metres throws results far off.