Radians–Degrees Conversion

Statement

Angles can be measured in two common units: degrees and radians. To convert between them, use:

\[ \text{radians} = \text{degrees} \times \frac{\pi}{180}, \qquad \text{degrees} = \text{radians} \times \frac{180}{\pi}. \]

Here, \(\pi \,\text{radians} = 180^\circ\). This equivalence allows conversion in either direction.

Why it’s true

• A full circle is \(360^\circ\) but also \(2\pi\) radians.
• Therefore, \(180^\circ = \pi\) radians, giving the conversion factors.
• This ratio ensures consistent angle measurement in trigonometry and calculus.

Recipe (how to use it)

1. If converting degrees to radians, multiply by \(\pi/180\).
2. If converting radians to degrees, multiply by \(180/\pi\).
3. Simplify results, leaving exact multiples of \(\pi\) when possible.

Spotting it

Whenever angles appear in trigonometric functions, check whether the question expects degrees or radians. Calculators often have a “mode” setting, so conversions are essential.

Common pairings

• Trigonometric values (sine, cosine, tangent) in radians vs degrees.
• Arc length and sector area formulas, which require radians.
• Calculus, where all angle measures are in radians.

Mini examples

1. \(90^\circ = 90 \times \pi/180 = \pi/2 \,\text{radians}\).
2. \(\pi/6 \,\text{radians} = (\pi/6) \times 180/\pi = 30^\circ\).

Pitfalls

• Forgetting to multiply by the right fraction (e.g. using \(180/\pi\) instead of \(\pi/180\)).
• Switching calculator mode incorrectly.
• Not simplifying exact results with \(\pi\).

Exam strategy

• Always check whether your final answer should be in degrees or radians.
• If the question involves arc length or calculus, radians are required.
• Write results exactly in terms of \(\pi\) when appropriate.

Summary

The degree–radian conversion is a core skill in GCSE and beyond. By remembering that \(180^\circ = \pi\) radians, you can confidently switch between units and apply the correct form for trigonometry, geometry, and calculus problems.