Degrees or Radians? Why Your Calculator Says sin 30 = -0.988

Type sin(30) into a calculator expecting 0.5, and the screen shows something like -0.9880316241 instead. Nothing is broken. The calculator is in radians mode, not degrees, so it worked out the sine of 30 radians instead of 30 degrees. Switch the angle unit back to degrees and the same keys give you 0.5. This page shows exactly where that degrees or radians setting lives on a calculator, how to switch it on a handheld and on WideMath, how to convert between the two by hand, and when radians are genuinely the unit you want.

Why your calculator says sin 30 = -0.988

Your calculator is reading the 30 you typed as 30 radians instead of 30 degrees. One radian is about 57.3 degrees, so 30 radians is almost five full turns around the circle before the sine function even gets to work. The angle unit, not a broken calculator, is the entire problem.

The number itself is not random or a rounding glitch. The sine of exactly 30 radians works out to -0.9880316241, and it will show the same figure on any calculator that is set to radians, every time. Switch to degrees and sin(30) drops back to the familiar 0.5, or on WideMath, the exact fraction it stands for:

sin(30) = 1/2

WideMath keeps exact answers instead of decimals wherever it can, so a fresh sin(30) in DEG mode prints 1/2, not 0.5. Tap the FRAC pill if you want to see it as a decimal, they are the same number. That habit of keeping fractions exact instead of rounding them off is explained further in how fractions stay exact on WideMath.

Degrees, radians or gradians: how to tell your calculator's mode

Every scientific calculator keeps the current angle unit on screen somewhere, and the habit worth building is checking it before you press a trig key, not after you distrust the answer. On a handheld, look for a small D, R or G printed near the top of the display, usually beside the memory and shift indicators. It is easy to miss because it is tiny, and it is exactly the letter that would have caught the -0.988 above before it happened.

WideMath shows the same letter in the same place: a single D, R or G sits at the right edge of the thin status strip above the display, next to the S, A and M indicators a handheld also prints there. Just under the display itself, a pill button spells the same setting out in full, DEG, RAD or GRAD, so you do not have to remember what the letter means. Both update together. There is only one angle unit active at a time, and it applies to every trig key on the keypad, sin, cos, tan and their inverses, not only the one you happen to be using.

How to change your calculator from radians to degrees

On most handhelds you switch the unit through a SETUP or MODE menu: press SHIFT then MODE, and pick Deg, Rad or Gra from the short list that opens, or press a dedicated DRG key if the model has one. There is no single universal keystroke for this, because manufacturers do not agree on where to put it, which is a real reason the mistake is so common: the setting sits one menu deep, and it quietly stays wherever the last person, or the last exam, left it.

On WideMath, tap the DEG pill under the display and it cycles forward one step at a time: DEG, then RAD, then GRAD, then back to DEG. If you want to jump straight to a specific unit, press SHIFT then MODE for the same SETUP list, which reminds you what a right angle looks like in each one before you commit: Deg reads sin(90) = 1, Rad reads sin(pi/2) = 1, and Gra reads sin(100) = 1. If you already have a number on the screen and want it converted to the next unit without retyping it, SHIFT then Ans runs that conversion directly on the displayed result. SHIFT is the same yellow-function key used everywhere else on the keypad; our walkthrough of the whole keypad covers what else it unlocks.

Converting degrees and radians by hand

You do not need a calculator for this. Degrees to radians: multiply by pi, then divide by 180. Radians to degrees: multiply by 180, then divide by pi. Both formulas come from one fact, that a full circle is 360 degrees and also 2 pi radians, so a straight line, half of that, is 180 degrees and pi radians.

30 degrees, the angle that started this page, converts to a clean fraction of pi:

30 × π ÷ 180 = π/6

Going the other way, 2 radians (a little under a third of a full turn) lands at an ordinary decimal, since it is not a clean multiple of anything:

2 × 180 ÷ π = 114.591559

Neither answer depends on which angle unit the calculator happens to be set to when you type it. Multiplying and dividing are plain arithmetic, not trig, so the angle unit only matters once you feed a result into sin, cos or tan.

Quick sanity checks before you trust the result

Two results are worth memorising, because between them they catch almost every angle unit mistake in a couple of seconds. In degrees, sin(90) is exactly 1, and sin(30) is exactly 0.5. If a trig function comes back with a long, odd decimal instead, the calculator is very likely not in degrees.

sin(90) = 1 tan(45) = 1

Here is the same input, 30, run through sine in all three units, each shown to 10 significant digits, the precision WideMath's display uses for a result that is not a clean fraction:

The DEG row is the only one that answers "what is the sine of a 30 degree angle". The other two are correct answers to a different question, the sine of 30 radians and the sine of 30 gradians, worked out exactly and rounded the same way. On WideMath, the DEG cell above actually prints as the exact fraction 1/2, as shown earlier, not the decimal 0.5, unless you tap FRAC first. It is the same kind of quick check behind six tests that expose a broken calculator: know the right answer before you ask for it.

The same mistake with inverse trig

Inverse trig functions, sin⁻¹, cos⁻¹ and tan⁻¹ (SHIFT+sin, SHIFT+cos and SHIFT+tan on most keypads), fail the same way, and they are easier to miss because the input is just a plain number with no obvious angle in it to make you suspicious. Type sin⁻¹(0.5) in degrees and you get the angle back, 30:

sin⁻¹(0.5) = 30

Type the exact same thing in radians and you get 0.5235987756, which is not a wrong answer, it is a right answer measured a different way: 30 degrees and 0.5235987756 radians are the same angle, and that decimal is exactly pi divided by 6. The trap is reading it as a broken version of 30, when it actually answered a question you did not mean to ask.

When radians are the right choice

Degrees are for people: the corner of a room, a compass bearing. Radians are for anywhere angle and distance need to relate to each other directly, which covers most of calculus and a good part of physics.

Take the derivative of sin(x) in calculus, and the clean textbook result, cos(x), only holds when x is measured in radians. Work it out in degrees and a leftover factor of pi divided by 180 shows up in front of cos(x), because the limit the whole proof rests on, sin(x) divided by x approaching 1 as x approaches 0, is only true when x is in radians.

Angular speed works the same way. Something spinning at an angular speed of omega radians per second sweeps out omega metres of arc every second for every metre of radius, because a radian is defined so that arc length equals radius times angle, with no extra conversion factor in between. Write the identical speed in degrees per second and that direct relationship needs a conversion constant bolted on to keep working.

What is a gradian

A gradian, sometimes called a grad or a gon, splits a full circle into 400 equal parts instead of 360, so a right angle is exactly 100 gradians. It comes out of the metric reforms of 18th century France, the same instinct that gave the world the metre and the kilogram, applied to angles: pick a round, decimal number for a right angle and build everything else from it.

It never caught on the way degrees did, but it survives in surveying and some engineering software, where a round 100 for a right angle keeps certain calculations tidy. WideMath's GRAD mode exists mainly so a problem written in gradians, surveying coursework, for example, does not need converting to degrees first. If you never meet the word gradian in your own work, it is safe to leave the pill on DEG and never touch GRAD at all; the full documentation lists every mode WideMath supports if you are curious what else is in there.

Sources

• NIST, Definitions of the Units Radian, Neper, Bel, and Decibel, for the formal SI definition of the radian as arc length divided by radius.
• Wikipedia, Gradian, for the 400-gradians-per-circle definition and its origin in French metrication.
• OpenStax, Calculus Volume 1, section 3.5, for why the derivative of sin(x) is cos(x) only when x is in radians.
• Mathematics LibreTexts, Velocity and Angular Velocity, for the radians-per-second definition of angular speed.

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