Three ratios, one right triangle
Trigonometry starts with a single idea: in a right triangle, once you fix one of the non-right angles, the ratios between the sides are locked in. Those ratios are sine, cosine and tangent.
Because the ratios depend only on the angle, they are tabulated once and reused everywhere — from surveying and navigation to the crosswind component an aircraft feels on landing.
A worked angle
Take 40°. Feeding it through each function:
| Function | Value |
|---|---|
| sin 40° | 0.643 |
| cos 40° | 0.766 |
| tan 40° | 0.839 |
And running it backwards, arcsin(0.643) returns 40.0°, which is the check that the inverse really does undo the forward function.
Going backwards with the inverses
Most real problems give you the sides and want the angle, which is the inverse direction. Knowing how far you climbed and how far you travelled horizontally, arctan of the ratio gives the slope angle. The one catch is that inverses return a single principal value, so an angle outside the standard band needs a full turn added or removed to find.
Frequently asked questions
What do sine, cosine and tangent mean?
In a right triangle they are ratios of side lengths: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. The mnemonic SOH-CAH-TOA packs all three. For a 40° angle they come out as sin 0.643, cos 0.766 and tan 0.839.
What is the difference between degrees and radians?
They are two units for the same thing. A full turn is 360 degrees or 2π radians, so 180° = π radians ≈ 3.1416. Degrees are friendlier for everyday angles; radians are what calculus and most programming languages expect. This calculator switches between them without you converting by hand.
What do the inverse functions do?
They run the ratio backwards to recover the angle. If sin of an angle is 0.643, then arcsin(0.643) = 40°. Arccos and arctan work the same way for cosine and tangent. They are the tool you reach for when you know the sides and want the angle.
Why is tangent undefined at 90 degrees?
Because tangent is sine divided by cosine, and cosine is zero at 90°. Dividing by zero has no answer, so the value shoots off to infinity as you approach 90° from either side. The calculator leaves the field blank there instead of printing a number that would only mislead.
Do the inverse functions have more than one answer?
Yes, infinitely many, because the ratios repeat every full turn. Calculators return the single principal value — for arcsin and arctan between −90° and 90°, for arccos between 0° and 180°. If your problem has an angle outside that band, add or subtract a full turn to find it.