Two ways to measure the same angle
A degree divides one full rotation around a circle into 360 equal parts, a convention that traces back to ancient Babylonian astronomy and its base-60 number system. A right angle is 90 degrees, a straight line is 180 degrees, and a full turn is 360 degrees.
A radian instead measures an angle by the ratio of the arc length it cuts off on a circle to that circle's radius: an angle of 1 radian is the angle for which the arc length exactly equals the radius. Because a circle's full circumference is 2 x pi x radius, a full rotation measures exactly 2 x pi radians -- approximately 6.2832 radians -- regardless of the circle's actual size, since radians are a ratio of two lengths and therefore have no units of their own.
Why radians are the natural unit for calculus
Radians are defined directly from the geometry of the circle itself (arc length divided by radius), rather than from an arbitrary division like 360, which is why formulas expressed in radians tend to come out simpler. The arc length formula, for example, is simply s = r x theta when theta is in radians, with no extra conversion constant needed; the same formula in degrees requires an additional factor of pi/180.
This simplicity becomes essential in calculus: the derivative of sin(x) is exactly cos(x) only when x is measured in radians. If x were measured in degrees instead, the derivative would pick up an extra constant factor of pi/180, since the degree-based sine function is really sin(x x pi/180) in terms of the radian-based function. Because this extra constant complicates every derivative, integral and series expansion involving trigonometric functions, mathematicians and scientists default to radians throughout calculus and higher mathematics, reserving degrees mainly for everyday and applied contexts like navigation, surveying and construction.
Converting between degrees and radians
Since a full rotation equals both 360 degrees and 2 x pi radians, the two units are related by a fixed ratio: to convert degrees to radians, multiply by pi/180; to convert radians to degrees, multiply by 180/pi. The table below lists verified conversions for the angles most frequently used in trigonometry.
| Degrees | Radians (exact, in terms of pi) | Radians (decimal) |
|---|---|---|
| 0° | 0 | 0 |
| 30° | pi/6 | 0.5236 |
| 45° | pi/4 | 0.7854 |
| 60° | pi/3 | 1.0472 |
| 90° | pi/2 | 1.5708 |
| 180° | pi | 3.1416 |
| 270° | 3pi/2 | 4.7124 |
| 360° | 2pi | 6.2832 |
The classic calculator-mode mistake
The single most common trigonometry error is entering an angle in one unit while a calculator or program is set to interpret it in the other, since both degrees and radians accept plain numeric input and neither mode produces an error message when the wrong one is selected -- only a wrong-looking answer that is easy to miss if the correct result is not already known.
For example, sin(30) evaluates very differently depending on the active mode: with the calculator set to degrees, sin(30 degrees) = 0.5 exactly. With the same calculator mistakenly left in radians mode, sin(30 radians) is approximately -0.9880 instead -- a value with the wrong sign and a wildly different magnitude, despite starting from the identical keystrokes. Because 30 radians corresponds to an angle of roughly 1,718.9 degrees (many full rotations past the intended 30-degree angle), the two results describe essentially unrelated angles.
Which unit should you use, and when
Degrees remain the standard for everyday and applied uses where the number itself needs to be human-readable and intuitive -- navigation headings, construction angles, geographic coordinates and most consumer calculators default to degrees for this reason. Radians are the standard in calculus, physics, engineering and any context involving derivatives, integrals or trigonometric series, because the underlying formulas are simpler and dimensionally cleaner in radians.
When switching between contexts -- for instance, taking a degree-based angle from a technical drawing and using it inside a calculus-based physics formula -- converting explicitly with the pi/180 or 180/pi factor before proceeding avoids the mode-mismatch error described above, and is safer than assuming a formula or programming function's default angle unit without checking its documentation.
Sıkça Sorulan Sorular
How do I convert degrees to radians?
Multiply the angle in degrees by pi/180. For example, 60 degrees x pi/180 = pi/3 radians, which is approximately 1.0472 radians.
How do I convert radians to degrees?
Multiply the angle in radians by 180/pi. For example, pi/4 radians x 180/pi = 45 degrees exactly.
Why does calculus use radians instead of degrees?
Because radians are defined directly from a circle's own geometry (arc length divided by radius), formulas involving trigonometric functions come out simpler in radians -- most importantly, the derivative of sin(x) equals exactly cos(x) only when x is in radians. In degrees, the same derivative picks up an extra constant factor of pi/180, complicating every derivative, integral and series expansion that follows.
What happens if you leave a calculator in the wrong angle mode?
The result will be numerically valid-looking but describe a completely different angle. For example, sin(30 degrees) = 0.5 exactly, while sin(30 radians) is approximately -0.9880, because 30 radians corresponds to roughly 1,718.9 degrees -- many full rotations past the intended 30-degree angle.
Is pi radians equal to 180 degrees?
Yes. A full rotation equals both 360 degrees and 2 x pi radians, so half a rotation -- a straight line -- equals both 180 degrees and exactly pi radians (approximately 3.1416 radians).
Kaynaklar
- Weisstein, Eric W. "Radian" and "Degree." MathWorld — A Wolfram Web Resource. mathworld.wolfram.com.
- Stewart J. Calculus. Cengage Learning (radian measure and derivatives of trigonometric functions).
- Larson R. Trigonometry. Cengage Learning (degree-radian conversion and angle measurement conventions).