Degrees vs radians: which mode to pick
Last reviewed on 30 April 2026.
Type sin(30) in the calculator with DEG selected and the answer is 0.5. Switch to RAD and the same expression returns roughly −0.988. Neither answer is a bug. They are the sine of a 30-degree angle and the sine of a 30-radian angle, two different angles in two different units. Picking the right mode is the single most common source of confusion when working with trigonometry on a calculator. This page explains what each unit measures, when to choose which, and how to convert when you need to.
What a degree measures
A degree slices a full turn into 360 equal parts. The number 360 is a leftover from ancient astronomy — it has many divisors, which makes it convenient for splitting a circle into halves, thirds, quarters, sixths, and so on without fractions. The right angle is 90°, a straight line is 180°, and a full turn is 360°. Degrees are the unit you grew up with: protractors, compass headings, latitude and longitude, and most everyday descriptions of angles use them.
What a radian measures
A radian is the angle subtended at the centre of a circle by an arc whose length equals the radius. That sounds abstract, so the practical way to remember it: a full turn is 2π radians, so 360° equals 2π rad and 180° equals π rad. One radian is roughly 57.2958°.
Radians are the natural unit for mathematics because they make the formulas of calculus simple. The derivative of sin(x) is cos(x) only when x is in radians; in degrees, an extra factor of π/180 sneaks in. Power series, Fourier analysis, complex exponentials, and most physics derivations assume radians without saying so.
When to use each
| Use degrees when | Use radians when |
|---|---|
| Reading an angle off a protractor or compass | Working through calculus exercises |
| Doing trigonometry from a geometry textbook | Applying Taylor series or differential equations |
| Surveying, navigation, or construction | Physics involving angular velocity or wave equations |
| The angle is given as a whole number with the ° symbol | The angle is given as a multiple of π |
| Working with sin, cos, tan in a calculator-only homework problem | Programming languages — most Math.sin functions take radians |
Quick conversion
Two formulas are enough for almost every situation:
radians = degrees × π / 180
degrees = radians × 180 / πSo 90° equals π/2 rad, 60° equals π/3 rad, and 1 rad equals roughly 57.2958°. The calculator on this site does not have a dedicated convert-to-radians key because the formula is short enough to type: enter 30 * pi / 180 and you will see 0.5235987... For the inverse, type 1.5 * 180 / pi to get about 85.94°.
A worked example: sine of 30, two ways
The cleanest demonstration of the difference is a single number computed both ways.
- DEG mode. Type
sin(30). The calculator interprets 30 as degrees, multiplies by π/180 to get π/6 radians, and returnssin(π/6) = 0.5. - RAD mode. Type the same
sin(30). Now 30 is treated as radians directly. Thirty radians is almost five full turns plus a remainder of about −1.42 rad, andsin(−1.42) ≈ −0.988.
Same keystrokes, very different answer. If your textbook expects 0.5, you are in DEG; if it expects something with π in it, you are probably in RAD.
How to pick the mode on the calculator
The two pills above the display set the mode. The active pill is highlighted; the inactive one is dim. Tap or click to switch. The mode applies to sin, cos, tan, and the inverse trig keys. Other functions such as log, sqrt, and e^x are unaffected — they have nothing to do with angle units.
The calculator starts in DEG because that matches what most users expect for a quick angle calculation. Once you switch to RAD, the choice persists for as long as the page is open. Reload and it goes back to DEG. See how to use the calculator for the full set of mode pills.
Common mistakes
- Forgetting which mode the page is in. The pill is small and easy to miss. If a result is far from your estimate, glance at the mode before retyping.
- Mixing units in the same expression. Writing
sin(30) + cos(π/6)in DEG mode treats 30 as degrees and π/6 as roughly 0.524 degrees, not π/6 radians. Either convert all angles or switch to RAD. - Calling a programming function with degrees. Almost every general-purpose programming language uses radians for trig functions. JavaScript, Python, C, Java, and Go all treat
sin(30)as 30 radians. Convert before passing the argument in. - Inverse trig and the principal value.
sin⁻¹(x)returns an angle in the range −90° to 90° (or −π/2 to π/2). If your problem expects an angle in the second or third quadrant, you may need to adjust by 180° or add 2π.
A short detour: gradians and turns
Some specialist calculators offer a third mode called "grad" or "gon", which divides a full turn into 400 parts. It is occasionally seen in surveying. There is also the "turns" unit, where a full circle is exactly 1. The calculator on this site does not include either, because the practical difference compared with degrees is small and the risk of accidental mode confusion is real. If you ever encounter a result that looks half right — for instance, an angle that is roughly your expected value times 9/10 — a stray gradian conversion is a candidate explanation.
Decision checklist
- Does the source angle have a degree symbol or come from a protractor? Use DEG.
- Is the angle given as a multiple of π? Use RAD.
- Is this a calculus or physics exercise that says nothing about units? Default to RAD.
- Are you converting from a programming language's
Math.sin? Use RAD. - Still unsure? Compute the value in both modes and compare against your rough estimate.