Degrees vs Radians vs Gradians: Angle Modes and Common Mistakes

The most common reason for wrong trigonometry answers on a calculator is not the formula but the angle mode. The same sin(30) is 0.5 in DEG and −0.988 in RAD. This guide covers the three angle units, how to convert between them, how to enter degrees in SciKey regardless of the mode, and why radians exist at all.

1. Three angle units

A radian is the central angle subtended by an arc as long as the radius. For a circle of radius r, arc length s and angle θ in radians are related by the simple formula s = rθ. A full circle has circumference 2πr, so a full turn is 2π radians.

2. Conversion formulas

• radians = degrees × π / 180
• degrees = radians × 180 / π
• 1 rad = 57.2957795131°, 1° = 0.0174532925199 rad
• 1 gon = 0.9°, 100 gon = 90°

SciKey has conversion functions: rad(30) = 0.523598775598 (= π/6) and deg(1) = 57.2957795131.

3. Wrong-mode errors

In RAD mode, sin(30) is the sine of 30 radians, about 4.77 full turns, so the answer is completely different. If a result is unexpectedly negative or just odd, check the mode indicator before doubting the formula.

The last row is a real problem: a building seen from 50 m away at an elevation angle of 35° is 50 tan 35° ≈ 35.01 m tall. RAD mode gives 23.69 m, which looks plausible and is therefore easy to miss. Two habits prevent this:

1. Set the mode before you start.
2. Attach ° to angles given in degrees.

4. The ° suffix: degrees in any mode

Appending ° to a number marks it as degrees and converts it to the current angle unit. So even in RAD mode, sin(30°) is 0.5. This lets you handle an occasional degree value in a mostly-radian physics problem without switching modes.

Going the other way, to feed a radian value into DEG mode, wrap it in deg(). Example in DEG mode: sin(deg(π/6)) = 0.5.

In DEG and GRAD modes, trig functions first reduce the argument exactly by 360° (400 gon) and return exact values at multiples of 90°. So sin(180) is 0, cos(90) is 0, and tan(90) is undefined and returns an error.

5. Units of inverse trig results

asin, acos and atan return results in the current angle unit.

Inverse trig functions only return values in a fixed range: asin and atan give −90° to 90°, acos gives 0° to 180°. That is why the direction of the point (−1, −1) computed as atan((-1)/(-1)) comes out as 45°, although the point is really in the third quadrant at −135° (or 225°). For directions from coordinates, use the quadrant-aware atan2(y, x): atan2(-1, -1) = −135 (DEG).

Inputs outside −1 to 1, as in asin(2), return a math error. Watch out for measured values like 1.0000001 that sneak past 1 after rounding.

6. When to use radians

Calculus

The derivative of sin x equals cos x only when x is in radians; in degrees a factor of π/180 appears. The small-angle approximation sin x ≈ x also assumes radians: sin(0.1) (RAD) = 0.0998334166, only 0.17% below 0.1. Taylor series, standard integrals and Fourier analysis all assume radians.

Physics: angular velocity and circular motion

Angular velocity ω is measured in rad/s. For a shaft turning at 1,500 rpm:

1. 1500×2π/60 → 157.079632679 (rad/s, = 50π)
2. Linear speed 0.2 m from the axis: Ans×0.2 → 31.4159265359 (m/s)

The formula v = rω works without any conversion factor precisely because ω is in radians. Oscillations (x = A sin ωt) and AC circuits (phase angles) are also computed in radians. Since ωt is already in radians, the calculator must be in RAD mode.

Arc length

For an arc of radius 5 m and central angle 72°, s = rθ needs θ in radians: 5×rad(72) → 6.28318530718 (m, = 2π). Plugging in degrees directly and getting 5×72 = 360 is a common slip.

7. Example: road grade (%) and angle

A road sign reading "10% grade" is not an angle. It means the road rises 10 m for every 100 m horizontally. To convert to an angle, use inverse tangent: in DEG mode, atan(0.1) → 5.7105931375°. Going the other way, a 3° slope is 100tan(3) → 5.24077792830 %. Slopes given as a ratio, such as a 1:12 ramp, work the same way: atan(1/12) → 4.76364169072°. In RAD mode atan(0.1) returns 0.0996686524912, so if the answer is suspiciously close to 0.1, suspect the mode.

8. A ten-second mode check

If you missed the indicator, evaluate sin(90). DEG gives exactly 1, RAD gives 0.893996663601, and GRAD gives 0.987688340595. The results are distinct enough to make this a quick check during an exam.

9. Summary

• A ° symbol in the problem means DEG; π, rad/s or ωt means RAD.
• If in doubt, attach the unit directly, as in 30°.
• Inverse trig results come out in the current unit; use atan2 when the quadrant matters.
• Asking yourself "should this sine really be negative?" is the most reliable check of all.

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Last updated: 2026-09-24