Radian Measure

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 $π$
- $2 π$ 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

1 radian shown on a unit circle

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:
- $2 π rad = 360 °$
- $π rad = 180 °$
- $\frac{π}{2} rad = 90 °$
- $\frac{π}{3} rad = 60 °$
- $\frac{π}{4} rad = 45 °$
- $\frac{π}{6} rad = 30 °$
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

$using multiples or fractions of pi when converting between radians and degrees$

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

(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{73 π}{300} = 0.764454 . . .

Note that

\frac{73 π}{300}

is the exact value answer

0.764 rad (3 s.f.)

(b)

Convert

\frac{5 π}{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{5 π}{4} \times \frac{180}{π} = \frac{5}{4} \times 180 = 5 \times 45 = 225

225°

Radian Measure (Edexcel IGCSE Further Maths)