Why sin(30) gives −0.988: degrees vs radians on a calculator
If sin(30) does not give 0.5, your calculator is in radian mode. How to tell, how to switch, how to convert between degrees and radians, and when to use each.
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Tool for this job: Scientific Calculator
You type sin(30) expecting 0.5 and the calculator says −0.988. Or cos(60) gives −0.952 instead of 0.5. The calculator is not broken: it is measuring angles in radians, and 30 radians is a very different angle from 30 degrees.
Two ways to measure an angle
- Degrees: a full turn is 360°. Used in everyday geometry, navigation and most school problems.
- Radians: a full turn is 2π, about 6.283. Used in calculus, physics and programming, because the formulas are simpler in radians.
180° = π radians. So 30° is π ÷ 6, about 0.524 radians, while 30 radians is almost five full turns, which lands at an angle whose sine is −0.988.
How to tell which mode you are in
Test with a known value: sin(90) should be 1 in degrees. If you get 0.894, you are in radians. Most calculators show DEG or RAD on the display. The Scientific Calculator has DEG and RAD buttons; the highlighted one is active.
Converting
- Degrees to radians: multiply by π ÷ 180. 45° × π ÷ 180 = 0.785.
- Radians to degrees: multiply by 180 ÷ π. 1 radian ≈ 57.3°.
| Degrees | Radians | sin | cos |
|---|---|---|---|
| 0° | 0 | 0 | 1 |
| 30° | π/6 | 0.5 | 0.866 |
| 45° | π/4 | 0.707 | 0.707 |
| 60° | π/3 | 0.866 | 0.5 |
| 90° | π/2 | 1 | 0 |
| 180° | π | 0 | −1 |
Gradians
Some calculators have a third mode, GRAD or gradians, where a right angle is 100 and a full turn 400. It is used in some surveying work and almost nowhere else. If a calculator shows GRAD, switch it to DEG or RAD before doing anything else; sin(30) in gradians gives about 0.454, another plausible-looking wrong answer.
Inverse functions too
asin, acos and atan return an angle, in whichever mode is active. asin(0.5) gives 30 in degree mode and 0.524 in radian mode. If an answer looks far too small, check the mode before reworking the problem.
Exams and homework
- If the question uses the degree symbol or comes from geometry, surveying or navigation, use degrees.
- If angles are written with π, or the work involves calculus, oscillations or circular motion, use radians.
- Check the mode before every exam. It is one of the most common causes of lost marks.
In spreadsheets and code
Excel, Google Sheets and nearly every programming language expect radians. In a spreadsheet, =SIN(RADIANS(30)) returns 0.5, and =DEGREES(PI()) returns 180. In JavaScript, Python and similar languages, convert with angle × Math.PI ÷ 180 before calling sin or cos.
For the order in which a calculator evaluates longer expressions, see order of operations explained.