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AI HL · 12.10

Geometry & Trigonometry

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12.10.10

Radians and degrees

12.10.10

Radians and degrees

Target: I can say what one radian is: the angle at the centre when the arc is as long as the radius. (+4 more)

By the end of this section

Key points

What a radian is

  • One radian is the angle at the centre of a circle when the arc is as long as the radius. It is about 57.3°.
  • Angle in radians = arc length ÷ radius. The radius fits round the circle 2π times, so a full turn is 2π radians.
  • Radians have no unit symbol. Write \(\theta = 2.5\) or \(\theta = 2.5\) rad. Degrees always carry the ° sign.

Converting

  • Half a turn: \(180^\circ = \pi\) radians. Everything else follows from this.
  • Degrees to radians: multiply by \(\frac{\pi}{180}\). Radians to degrees: multiply by \(\frac{180}{\pi}\).
  • Standard angles: \(30^\circ=\frac{\pi}{6}\), \(45^\circ=\frac{\pi}{4}\), \(60^\circ=\frac{\pi}{3}\), \(90^\circ=\frac{\pi}{2}\), \(180^\circ=\pi\), \(270^\circ=\frac{3\pi}{2}\), \(360^\circ=2\pi\). Learn them.
  • Give an exact answer as a fraction of π, in lowest terms. Use a decimal (3 significant figures) only when the question asks for one or the angle has no π in it.

Negative angles and large angles

  • A negative angle is measured clockwise. Keep the sign when you convert.
  • An angle bigger than 360° (or 2π) is more than one turn. Convert it as it is: do not take off whole turns unless asked.
  • Set the calculator to the right angle mode before any sine, cosine or tangent. A radian angle in degree mode gives a wrong value without any warning.

Explore: what one radian is

Drag the point round the circle. The arc is measured in radius lengths. The number of radius lengths is the angle in radians.

θ = arc ÷ radius =

Arc length
Angle in radians
Angle in degrees
As a multiple of π

Drag the point round the circle.

  1. Press "1 radian". Is the angle a little more or a little less than 60°? How can you tell from the picture?
  2. Change the radius. What changes in the read-outs and what stays the same? Why does the angle in radians not depend on the radius?
  3. Press 2π. How many radius lengths go round the whole circle, and where have you seen that number before?
  4. Switch to multiples of π/12. What are π/6, π/4 and π/3 in degrees?

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
What one radian is

An arc of length 12.5 cm lies on a circle of radius 5 cm. (a) Find the angle at the centre in radians. (b) Write the angle in degrees, to 3 significant figures.

θ5 cm12.5 cmOAB
Working
  1. The rule

    \(\theta=\dfrac{\text{arc length}}{\text{radius}}\)

    One radian is the angle when the arc is the same length as the radius.

  2. Radians

    \(\theta=\dfrac{12.5}{5}=2.50\) rad

    Arc ÷ radius. The answer has no π.

  3. Degrees

    \(2.50\times\dfrac{180}{\pi}=143.2^\circ\) so \(143^\circ\)

    Radians to degrees: multiply by 180/π.

Video walkthrough coming
Completing a table of angles

Complete the table. Give the radians as multiples of π.

Degrees120135210?Radians???5π/3
Working
  1. Start from 180°

    \(180^\circ=\pi\)

    Use this to find the fraction of π for each angle.

  2. Each angle

    \(120^\circ=\frac{120}{180}\pi=\frac{2\pi}{3}\), \(135^\circ=\frac{3\pi}{4}\), \(210^\circ=\frac{7\pi}{6}\)

    Write the fraction of 180 and cancel.

  3. The last column

    \(\frac{5\pi}{3}\times\frac{180}{\pi}=300^\circ\)

    Radians to degrees: the π cancels.

  4. Degrees120135210300Radians2π/33π/47π/65π/3
    Completed table

    120° = 2π/3, 135° = 3π/4, 210° = 7π/6, 300° = 5π/3

    Check each with a standard angle: 135° is 90° + 45°, so π/2 + π/4 = 3π/4.

Video walkthrough coming
A negative angle over one turn

Write −400° in radians, in terms of π. Then give it as a decimal to 3 significant figures.

Working
  1. Multiply by π/180

    \(-400\times\frac{\pi}{180}\)

    Degrees to radians. Keep the minus sign.

  2. Cancel

    \(-\frac{400\pi}{180}=-\frac{20\pi}{9}\)

    Divide top and bottom by 20. Do not subtract a full turn.

  3. Decimal

    \(-\frac{20\pi}{9}=-6.98\)

    From the calculator, in 3 significant figures.

Video walkthrough coming
Radians to degrees

Convert \(-\frac{13\pi}{5}\) radians to degrees.

Working
  1. Multiply by 180/π

    \(-\frac{13\pi}{5}\times\frac{180}{\pi}\)

    Radians to degrees.

  2. The π cancels

    \(-\frac{13\times180}{5}=-13\times36\)

    180 ÷ 5 = 36.

  3. Answer

    \(-468^\circ\)

    More than one full turn clockwise: 468 = 360 + 108.

Video walkthrough coming
Radians without a π

An angle is 2.7 radians. Give it in degrees to 3 significant figures.

Working
  1. Multiply by 180/π

    \(2.7\times\frac{180}{\pi}\)

    There is no π in the angle, so the answer is a decimal.

  2. From the calculator

    \(154.7\ldots^\circ\)

    Check: 1 radian is about 57.3°, so 2.7 radians is a little under 160°.

  3. Round

    \(155^\circ\)

    3 significant figures.

Video walkthrough coming