Calculate principal values, explore domain restrictions on the unit circle, and reference standard angles.
Trigonometry functions (like sine and cosine) take an angle and give you a ratio. Inverse Trigonometry functions do the exact opposite: they take a ratio and give you back the angle.
This Inverse Trig Studio helps you reverse the process. Whether you are looking for an angle in a right triangle or solving complex equations in calculus, these tools provide the answers with visual context.
Imagine a function as a machine. You put in "30°", the Sine machine whirs, and out pops "0.5".
The Inverse Sine machine (Arcsin) runs in reverse. You put in "0.5", and out pops "30°".
Mathematically, if sin(x) = y, then arcsin(y) = x. They are mirror images of each other.
Here is the problem: Sin(30°) is 0.5. But Sin(150°) is also 0.5. And Sin(390°)...
If you ask "Which angle gives a sine of 0.5?", there are infinite answers!
To make Arcsin a real function (which must have only one output), we ignore all the other answers and focus only on the Principal Values. For Arcsin, we only look at angles between -90° and 90°.
This is the most common mistake in trigonometry.
The "-1" exponent here means "Inverse Function", not "One Divided By".
Builders use inverse trig every day. If you know the height of a roof (rise) and the width of the building (run), how do you cut the rafters at the correct angle?
You use Arctan(rise / run). This tells you exactly what angle to set your saw to.
The input for both arcsin(x) and arccos(x) must be between -1 and 1 inclusive [-1, 1]. You cannot have a triangle side longer than the hypotenuse, so sine/cosine can never exceed 1.
Tangent is opposite/adjacent. The opposite side can be infinitely large compared to the adjacent side (imagine a line going almost straight up). Therefore, the input ratio can be any number from negative infinity to positive infinity.
Look for the '2nd' or 'Shift' button, then press the Sin, Cos, or Tan button. You should see sin⁻¹ appear on the screen.
Because trig functions repeat (periodic), inverse trig functions have infinite possible answers. The Principal Value is the single, standardized answer chosen by mathematicians to be the calculator output (e.g., -90° to 90° for arcsin).
These are the inverses of the reciprocal functions: Cotangent, Secant, and Cosecant. They are rarely used in basic geometry but appear frequently in calculus integrals.
Yes! If it's a right triangle, use SOH CAH TOA. If it's not a right triangle, you can use the Law of Sines or Law of Cosines, which also involve inverse trig to solve for the angle.
No! arcsin(sin(x)) is only x if x is inside the restricted domain [-90°, 90°]. For example, arcsin(sin(150°)) = arcsin(0.5) = 30°, NOT 150°.
In calculus, the derivative of arcsin(x) is 1 / √(1 - x²). This is why inverse trig functions strangely appear when integrating algebraic fractions.
The notation was introduced by John Herschel in 1813. Before that, 'arc' notation (arcsin) was more common, meaning 'the arc whose sine is...'.
Radians make calculus formulas much simpler. If you use degrees, derivatives of trig functions get messy constants (like π/180). With radians, the derivative of sin(x) is just cos(x).