Degrees vs. Radians: When to Use Each Angle Mode

If you’ve ever typed sin(30) into a calculator and gotten -0.988 instead of 0.5, you’ve run into the single most common calculator mistake: the wrong angle mode. The good news is there’s no error message to decode — you just need to know which of the three modes to reach for.

The three modes

Degrees split a full turn into 360 pieces. This is the mode most people learn in school, and it’s the right choice for geometry, navigation, surveying, and most everyday trig problems (“a 30° angle,” “a 45-45-90 triangle”).

Radians measure an angle by the arc length it sweeps out on a unit circle, so a full turn is 2π radians. Radians are the native unit of calculus — derivatives and integrals of sin(x) and cos(x) only come out clean when x is in radians. If you’re differentiating, integrating, or working with any formula that already has a π baked into it, you’re almost certainly expected to work in radians.

Gradians split a full turn into 400 pieces, so a right angle is exactly 100 gradians. They’re rare outside of surveying and some European engineering contexts, but they exist for a reason: 100 is a much friendlier number to divide than 90.

Why this trips people up

Trig functions don’t know what unit you meant — they just apply the same formula to whatever number you give them. sin(30) in radian mode computes the sine of an angle nearly five full turns around the circle, which is why you get a plausible-looking but completely wrong decimal instead of a clean 0.5. There’s nothing to catch this for you; the calculator did exactly what you asked.

A quick way to sanity-check your mode

sin(90) should be exactly 1 in degree mode. sin(π/2) (typed with the actual pi constant) should be exactly 1 in radian mode. If neither gives you a clean 1, check your mode before you check your formula.

Try it yourself

The TI-84 Calculator keeps the current angle mode visible at all times in a small pill above the display — tap it to cycle through Deg → Rad → Grad. The same mode applies consistently across the scientific calculator, the graphing engine (including polar plots, where the angle mode changes what a full loop of θ looks like), and complex-number arguments, so you only ever have to think about it in one place.