Degrees vs radians: why your calculator gives the "wrong" answer

The quick fix

Look for DEG, RAD or D / R on your calculator's display. On Kalkuli it's the button above the display. If your angles are in degrees (they have a ° symbol, or they're things like 30, 45 and 90), set it to DEG.

A five-second check that works on any calculator: type sin(30). If you get 0.5, you're in degrees. If you get −0.988, you're in radians.

What a radian actually is

Degrees are arbitrary: the Babylonians split a circle into 360 parts, probably because 360 divides so neatly and is close to the number of days in a year. Radians come from the circle itself.

Take a circle and measure off an arc along its edge that's the same length as the radius. The angle at the centre is one radian. It's a bit over 57°.

Because the circumference is 2π times the radius, a full turn is 2π radians. That gives the conversions:

So when a calculator in radian mode sees sin(30), it's working out the sine of 30 radians. That's 30 × 57.2958 ≈ 1,718.9°, which is 4 full turns plus 278.9°, and sin 278.9° ≈ −0.988. The calculator is right; it's just answering a different question.

Why mathematicians bother with radians

Radians make the formulas cleaner. With θ in radians:

• Arc length is just s = rθ. A 40° arc on a circle of radius 0.5 m is 0.5 × 0.6981 = 0.349 m. In degrees you'd need s = πrθ/180.
• Sector area is ½r²θ.
• Small-angle approximation: for small θ, sin θ ≈ θ. sin(5°) = 0.08716 and 5° = 0.08727 rad, less than 0.2% apart. This is what makes pendulum and optics formulas simple, and it only works in radians.
• Calculus: the derivative of sin x is cos x only when x is in radians. In degrees it's (π/180) cos x, and that awkward factor would appear everywhere.

That's why anything involving calculus, oscillations, waves or angular velocity is written in radians, even when nobody says so.

Which mode do you need?

Use degrees when:

• the angle has a ° symbol, or is a "round" value like 30, 45, 60 or 120;
• you're solving a triangle, doing GCSE or A-level geometry, surveying, navigation or bearings;
• you're reading angles off a protractor, a drawing or a CAD package.

Use radians when:

• π appears in the angle (sin(π/6), cos(2πft));
• you're differentiating or integrating trig functions;
• you're working with angular frequency ω, oscillations, AC circuits or waves;
• the angle is a plain decimal with no units from a physics formula, such as θ = 0.35.

Where radians catch people out

Spreadsheets

Excel, Google Sheets and LibreOffice all use radians, and there's no mode to change. =SIN(30) gives −0.988. Use =SIN(RADIANS(30)), or =SIN(30*PI()/180). The inverse functions also return radians, so wrap them: =DEGREES(ASIN(0.5)) gives 30.

Programming languages

JavaScript's Math.sin, Python's math.sin, C's sin() and practically every other language work in radians only. Python has math.radians() and math.degrees() to convert. A surprising number of bugs in games and robotics code come from feeding degrees into these functions.

Revolutions per minute

Engineering formulas such as power = torque × angular velocity (P = Tω) need ω in radians per second. A motor spinning at 3,000 rpm is turning at

Forget the 2π and your power figure is 6.28 times too small.

Inverse functions

sin⁻¹, cos⁻¹ and tan⁻¹ return their answers in whichever mode the calculator is in. In radian mode, sin⁻¹(0.5) = 0.5236, which is π/6, and is correct; it just isn't in degrees.

Gradians

Some handheld calculators have a third mode, GRAD or G, where a right angle is 100 gradians. It's used in some surveying work in continental Europe and almost nowhere else. If your calculator shows G and your answers are slightly off, that's why: 30 gradians is only 27°.

Converting by hand

On Kalkuli you can do the conversion in either mode by typing the formula: 135 × π ÷ 180 or 2.5 × 180 ÷ π.