Two units for the same angle
Degrees divide a full rotation into 360 equal parts. Radians measure an angle by the ratio of intercepted arc length s to radius r: θ=s/r. A full circle has circumference 2πr, so its angle is 2π radians. Therefore 360°=2π rad, 180°=π rad, and 90°=π/2 rad. The angle is unchanged when its numerical measure is converted.
Radians are dimensionless ratios, though writing rad clarifies the unit. They connect angles directly to arc length and make many calculus formulas natural. Degrees remain common in geometry, navigation, and everyday descriptions. Neither unit is intrinsically more accurate. Error occurs when a numerical value is interpreted in the wrong unit or when a tool's mode is assumed rather than checked.
Conversion formulas
To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π. The factors equal one because π radians and 180 degrees represent the same straight angle. Keep π symbolic until the final display when an exact answer is useful. Cancelling common factors often gives a familiar angle immediately.
For 30°, compute 30×π/180=π/6 rad. For π radians, compute π×180/π=180°. A decimal radian input can be converted numerically, but the result inherits its precision. Do not attach a degree symbol to a radian number or write π as though it carried degrees by itself; the notation should identify the chosen measure.
Arc-length interpretation
Radian measure comes from θ=s/r, so arc length is s=rθ when θ is in radians. On a circle of radius 4, an angle of π/3 rad intercepts arc length 4π/3. If an angle is given in degrees, convert it before using this compact formula. Substituting 60 directly would exaggerate the arc because 60 is not the radian measure.
The ratio also explains why one radian is about 57.2958°. An arc equal in length to the radius subtends one radian. A larger radius creates a proportionally longer arc for the same central angle, leaving s/r unchanged. This geometric definition is independent of any calculator and provides a check on conversion direction.
NumUtility is degree-only
The NumUtility Scientific Calculator converts each sin, cos, or tan argument from degrees to radians internally before calling JavaScript's trigonometric functions. The interface has no angle-mode control. Enter sin(30) for the sine of 30 degrees. Do not enter sin(pi/6) expecting the radian interpretation; the calculator reads pi/6 as approximately 0.524 degrees.
If the source angle is 5π/6 radians, first convert: (5*pi/6)*180/pi=150 degrees, then evaluate sin(150). You may calculate the conversion as a separate expression and copy the degree result. Keeping conversion and trigonometric evaluation conceptually separate makes the mode explicit and avoids applying the 180/π factor twice.
Three verified angle examples
Input: 30 degrees. Rule: radians=degrees×π/180. Steps: 30×π/180, cancel 30 from 180. Result: π/6 rad. Verification: six angles of π/6 form π radians, or 180°.
Input: π radians. Rule: degrees=radians×180/π. Steps: π×180/π and cancel π. Result: 180°. Verification: both describe a straight angle and half of a full rotation.
Input: sin of π/6 radians using NumUtility. Rule: convert to degrees before the degree-only function. Steps: π/6×180/π=30; enter sin(30). Result: approximately 0.5. Verification: the special-angle value is exactly 1/2; entering sin(pi/6) directly instead evaluates about 0.524° and produces about 0.00914.
Tangent singularities and floating point
Sine and cosine are defined for every real angle, but tangent equals sin θ/cos θ and is undefined when cosine is zero. In degrees those singularities occur at 90°+180°k. The current calculator may show a huge finite number for tan(90) because floating-point π and the degree-to-radian conversion do not create an exact zero cosine. That display must not be interpreted as the mathematical tangent.
Floating-point effects also make sin(30) internally slightly different from exactly 0.5, although formatting normally shows 0.5. Use known exact special-angle values when an exact result is required. Near a tangent singularity, small input changes cause very large output changes, so a rounded angle can be especially misleading. This tool does not perform interval analysis or singularity detection.
Tables, formulas, and unit labels
A table should label its angle column ° or rad and should not mix units in unlabeled rows. An equation involving angular speed must state whether it uses degrees per second or radians per second. A graph's horizontal scale likewise changes when the unit changes: sine has period 360 in degrees and 2π in radians. The shape is the same after rescaling.
When checking a formula, inspect constants as well as inputs. Calculus identities such as d(sin x)/dx=cos x assume x is in radians; a degree-based derivative includes a π/180 factor. NumUtility evaluates numerical trigonometric values and does not differentiate expressions. Do not infer that its degree convention changes the underlying mathematical assumptions of a separate symbolic formula.
A safe conversion workflow
Identify the source unit from its label or context. Convert once using the correct reciprocal factor, retain π symbolically when possible, and estimate whether the result has the right size: radians are numerically smaller than degrees for the same ordinary positive angle. Then enter a degree value into NumUtility's trigonometric function. Record both the converted value and its unit.
Use the main scientific-calculator guide for supported syntax and the order guide for grouping conversion expressions. The calculator does not switch modes, accept a degree symbol, or infer radians from the presence of pi. Explicit conversion is required. This page explains mathematical angle units; it does not provide navigation, engineering, or scientific safety conclusions from an angle calculation.
Reference angles and repeated rotations
Coterminal angles differ by complete rotations: add or subtract 360° in degree measure or 2π in radian measure. Thus 390° and 30° have the same sine and cosine. Converting 390° gives 13π/6, which differs from π/6 by 2π. NumUtility accepts either degree value numerically, but reducing a large angle can make the geometry and expected sign easier to inspect.
Quadrants provide a sign check. Sine is positive in quadrants I and II, cosine in I and IV, and tangent where sine and cosine share a sign. For example, 150°=5π/6 lies in quadrant II, so sine is positive and cosine negative. These checks catch an accidental reciprocal conversion even when the calculator returns a plausible decimal.
Angle conversion does not convert other units in a formula. If angular speed is radians per second, multiplying only the angle by 180/π yields degrees per second; the time unit remains seconds. If a formula expects angular frequency in radians per second, feeding degrees per second without conversion changes the result. Preserve compound-unit labels and apply the factor to the angular component only.
A conversion check can use proportions: degree value/180 must equal radian value/π. For 45° and π/4, both ratios equal 1/4. This comparison avoids decimal approximation and exposes a swapped factor. When a radian value is already a decimal approximation, reversing the conversion should reproduce the original only to its stated precision. Do not replace a rounded radian input with a nearby exact multiple of π unless the source identifies that exact angle.