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CIE A-Level Maths Study Notes

1.4.1 Understanding Radians

Radians are a unit of angle measurement, similar to degrees. They offer a way to measure angles based on the radius of a circle. One radian is defined as the angle at the centre of a circle, where the length of the arc is equal to the radius of the circle.

Comparison with Degrees

  • Relation with Degrees: π radians are equivalent to 180°. This means that a full circle, which is 360°, is equal to 2π radians.
  • Conversion Formulas:
    • To convert radians to degrees: Multiply by 180π\frac{180}{\pi}.
    • To convert degrees to radians: Multiply by π180\frac{\pi}{180}.
radian diagram

Image courtesy of Mometrix

Practical Examples

Example 1: Converting Radians to Degrees

Convert 3π4\frac{3\pi}{4} radians into degrees.


$ \frac{3\pi}{4} = \frac{3\pi}{4} \times \frac{180}{\pi} = 3 \times \frac{180}{4} = 135° <h3><strong>Example2:ConvertingDegreestoRadians</strong></h3><p>Convert45°intoradians.</p><p><strong>Solution:</strong></p><h3><strong>Example 2: Converting Degrees to Radians</strong></h3><p>Convert 45° into radians.</p><p><strong>Solution:</strong></p>45° = 45 \times \frac{\pi}{180} = \frac{\pi}{4}<h2id="practiceexercises"><strong>PracticeExercises</strong></h2><h4><strong>1.Convert60°toRadians</strong></h4><p><strong>Solution:</strong></p><p>Toconvertdegreestoradians,usetheformula<h2 id="practice-exercises"><strong>Practice Exercises</strong></h2><h4><strong>1. Convert 60° to Radians</strong></h4><p><strong>Solution:</strong></p><p>To convert degrees to radians, use the formula \text{radians} = \text{degrees} \times \frac{\pi}{180}.</p>.</p>60° = 60 \times \frac{\pi}{180} = \frac{60}{180} \times \pi = \frac{1}{3} \times \pi = \frac{\pi}{3}<p>Therefore,60°isequalto<p>Therefore, 60° is equal to \frac{\pi}{3}radians.</p><h4></h4><h4><strong>2.Convert</strong> radians.</p><h4></h4><h4><strong>2. Convert </strong>(\frac{\pi}{3})<strong>RadianstoDegrees</strong></h4><p><strong>Solution:</strong></p><p>Toconvertradianstodegrees,usetheformula<strong> Radians to Degrees</strong></h4><p><strong>Solution:</strong></p><p>To convert radians to degrees, use the formula \text{degrees} = \text{radians} \times \frac{180}{\pi}.</p></p>\frac{\pi}{3} = \frac{\pi}{3} \times \frac{180}{\pi} = \frac{1}{3} \times 180 = 60°<p>Therefore,<p>Therefore, \frac{\pi}{3}radiansisequalto60°.</p><h4></h4><h4><strong>3.Convert120°toRadians</strong></h4><p><strong>Solution:</strong></p><p>Usethesameformulaforconvertingdegreestoradians.</p> radians is equal to 60°.</p><h4></h4><h4><strong>3. Convert 120° to Radians</strong></h4><p><strong>Solution:</strong></p><p>Use the same formula for converting degrees to radians.</p>120° = 120 \times \frac{\pi}{180} = \frac{120}{180} \times \pi = \frac{2}{3} \times \pi = \frac{2\pi}{3}<p>Therefore,120°isequalto<p>Therefore, 120° is equal to \frac{2\pi}{3}radians.</p><h4></h4><h4><strong>4.Convert</strong> radians.</p><h4></h4><h4><strong>4. Convert </strong>\frac{5\pi}{6}<strong>RadianstoDegrees</strong></h4><p><strong>Solution:</strong></p><p>Again,usetheformulaforconvertingradianstodegrees.</p><strong> Radians to Degrees</strong></h4><p><strong>Solution:</strong></p><p>Again, use the formula for converting radians to degrees.</p>\frac{5\pi}{6} = \frac{5\pi}{6} \times \frac{180}{\pi} = \frac{5}{6} \times 180 = 150°<p>Therefore,<p>Therefore, \frac{5\pi}{6}$ radians is equal to 150°.

Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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