Degrees vs radians: why sin 30 gives −0.988 on your calculator
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Why a calculator returns sin 30 = −0.988 instead of 0.5, what degrees and radians mean, the conversion formulas, a table of common angles and when to use each unit.
If sin 30 gives −0.988… instead of 0.5, the calculator is in radian (RAD) mode and read 30 as 30 radians. Tap the angle indicator at the top of the screen to switch to DEG and you get 0.5. To convert, radians = degrees × π/180.
What a radian is
A degree is 1/360 of a full turn. A radian is the angle made by an arc as long as the radius. A full circle’s arc is 2πr long, so a full turn is 2π radians.
- 360° = 2π rad, 180° = π rad
- 1 rad = 180°/π ≈ 57.29577951°
- Degrees → radians: × π/180. Radians → degrees: × 180/π.
Common angles
| Degrees | Radians | sin | cos |
|---|---|---|---|
| 0° | 0 | 0 | 1 |
| 30° | π/6 | 1/2 | √3/2 |
| 45° | π/4 | √2/2 | √2/2 |
| 60° | π/3 | √3/2 | 1/2 |
| 90° | π/2 | 1 | 0 |
| 180° | π | 0 | −1 |
| 360° | 2π | 0 | 1 |
This calculator returns these values exactly, as in the table. Tap S⇔D for the decimal (for example √3/2 → 0.8660254038).
When to use which
- DEG: triangles, geometry, surveying, slopes and bearings, anywhere you measure angles with a protractor.
- RAD: calculus, arc length (s = rθ), angular velocity, oscillations and waves. The rule that the derivative of sin x is cos x only holds in radians.
- GRA (gradians): a right angle is 100 gradians. Used in some surveying; you will rarely need it.
The SIN and COS functions in Excel and in programming languages take radians. In Excel, the sine of 30° is =SIN(RADIANS(30)).
Changing the mode here
- Tap DEG, RAD or GRA at the top left of the screen to cycle through them, or choose in Settings.
- Enter degrees-minutes-seconds with the °′″ key (SHIFT + bracket). 30°15′ is 30.25°.
- To see a result in degrees-minutes-seconds, tap °′″ under the result.
Found a mistake, or a rule that has changed? Email contact@lumenlab.page. We check it, fix it and put the new date on this guide.