Radian Measure (Edexcel IGCSE Further Maths)

Spec point — 10A

Radian Measure

What are radians?

• Radians are an alternative to degrees for measuring angles

• 1 radian is defined as the angle in a circle sector of radius 1 and arc length 1

• Radians are often given in terms of $π$

  • $2π$ radians = 360°
  • $π$ radians = 180°

• The symbol for radians is ^c but it is more usual to see rad

  • When $π$ is involved, no symbol for radians is needed

    • With $π$ it is obvious it is in radians

  • You should never omit the symbol for degrees

• In the exam a question will usually tell you which angle measure to use

  • But radians must be used if calculus with the trigonometric functions is involved

How do I convert between radians and degrees?

• Use $π$* *radians = 180° to convert between radians and degrees

  • To convert from radians to degrees multiply by $\frac{180}{π}$
  • To convert from degrees to radians multiply by $\frac{π}{180}$

• Some of the common equivalents are:

  • $2π\mathrm{rad}=360^{\circ}$
  • $π\mathrm{rad}=180^{\circ}$
  • $\frac{π}{2}\mathrm{rad}=90^{\circ}$
  • $\frac{π}{3}\mathrm{rad}=60^{\circ}$
  • $\frac{π}{4}\mathrm{rad}=45^{\circ}$
  • $\frac{π}{6}\mathrm{rad}=30^{\circ}$

• It is useful to remember these and use them to work out other conversions

• Your calculator will be able to work with both radians and degrees

Exam Hint

• If the question specifies a unit of angle measure (radians or degrees) be sure to use it!
• Make sure that your calculator is in the correct mode for the angle measure you are working with

Worked Example (a) Convert 43.8° to radians, giving your answer correct to 3 significant figures.

Multiply by $\frac{π}{180}$

$43.8\times\frac{π}{180}=\frac{73π}{300}=0.764454...$

Note that $\frac{73π}{300}$ is the exact value answer

Final answer: 0.764 rad (3 s.f.)

(b) Convert $\frac{5π}{4}$ radians to degrees.

If you remember that $\frac{π}{4}$ radians is 45°, then you just need to multiply that by 5 to get the answer.

Otherwise, multiply by $\frac{180}{π}$

$\frac{5π}{4}\times\frac{180}{π}=\frac{5}{4}\times180=5\times45=225$

Final answer: 225°