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Precalculus · F.TF.A.1 · G.C.B.5

Radian measure

Degrees cut a full turn into 360 parts for no reason found inside the circle. Can an angle be measured with the circle itself?

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Sectors, segments and their areasGeometryMeasuring anglesGrades 4 to 8
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Angular and linear speedPrecalculusArc length and sector area in radiansPrecalculusTrig functions on the unit circlePrecalculus
Where you are 4 in Angles and their measure
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Angles and their measure4
Angles in standard position and coterminal angles
Radian measure
Angular and linear speedArc length and sector area in radians

Why this exists

A degree is \displaystyle \frac{1}{360} of a full turn, and the number 360 is a choice made long ago, not something the circle forces. It works well for drawing and for naming angles, but it puts a stray factor into every formula that joins an angle to a length. In arc length the arc of an n^\circ angle is \displaystyle s = \frac{\pi n r}{180}, and that \displaystyle \frac{\pi}{180} is there only because degrees were chosen.

Radian measure removes it. It measures an angle by the one length the circle already has, its radius: how many radii of arc does the angle cut off? With that unit the arc is simply the radius times the angle, and the formulas of trigonometry and, later, calculus take their simplest form.

Degrees stay in use, so you need both and the bridge between them. Degrees also have their own finer units, minutes and seconds, used wherever an angle has to be read precisely: a surveyor’s instrument, a map’s latitude, or the joint of a robotic arm. This lesson covers all three.

Definition

Degrees, minutes and seconds. One degree is split into 60 minutes, and one minute into 60 seconds:

1^\circ = 60' \qquad 1' = 60'' \qquad 1^\circ = 3600''

An angle written 47^\circ 21' 36'' means 47 degrees plus 21 minutes plus 36 seconds. These minutes and seconds measure angle, not time; they only borrow the same base of 60.

Radian measure. Put the vertex of an angle at the center of a circle of radius r. The angle cuts off an arc of length s. The radian measure of the angle is the arc length divided by the radius:

\theta = \frac{s}{r}

One radian is the central angle whose arc is exactly one radius long, s = r.

1\text{ rad}
s = r
r
r
O
Figure 1One radian is the central angle whose arc is exactly as long as the radius.

Because s and r are both lengths in the same unit, the units cancel and a radian measure is a plain number. That is why an angle written with no degree sign, such as \theta = 2, is read as 2 radians. The word “radians” or “rad” may be added for clarity, but a degree measure must always carry its ^\circ.

Intuition

Take a string as long as the radius and lay it around the rim, end to end. Each length of string covers one radian. Since the circumference is 2\pi r, the rim holds 2\pi \approx 6.28 radii: six whole strings and a little over a quarter of a seventh.

0
1
2
3
4
5
6
r
r
O
Figure 2Six radii of arc fit around the rim with about 0.28 of a radius left over, because a full turn is 2\pi \approx 6.28 radians.

So a full turn is 2\pi radians, whatever the size of the circle. A bigger circle needs a longer string, but the string is still one radius, and it still fits around 2\pi times. That is what makes the radian a measure of the angle and not of the circle.

Results

The bridge between the units. A full turn is 360^\circ and also 2\pi radians, so

\pi \text{ rad} = 180^\circ

Dividing both sides gives the two conversion factors:

\begin{aligned} 1^\circ &= \frac{\pi}{180} \text{ rad} \approx 0.01745 \text{ rad} \\ 1 \text{ rad} &= \frac{180^\circ}{\pi} \approx 57.30^\circ \end{aligned}

To turn degrees into radians, multiply by \displaystyle \frac{\pi}{180}; to turn radians into degrees, multiply by \displaystyle \frac{180}{\pi}. Each factor equals 1, because its top and bottom name the same angle, so multiplying by it changes the unit and not the angle. This is the same move as a conversion factor between feet and inches.

Minutes and seconds to decimal degrees.

d^\circ\, m'\, s'' = \left(d + \frac{m}{60} + \frac{s}{3600}\right)^\circ

Common angles. Fractions of 180^\circ become the same fractions of \pi:

DegreesRadians
30^\circ\displaystyle \frac{\pi}{6}
45^\circ\displaystyle \frac{\pi}{4}
60^\circ\displaystyle \frac{\pi}{3}
90^\circ\displaystyle \frac{\pi}{2}
120^\circ\displaystyle \frac{2\pi}{3}
135^\circ\displaystyle \frac{3\pi}{4}
150^\circ\displaystyle \frac{5\pi}{6}
180^\circ\pi
270^\circ\displaystyle \frac{3\pi}{2}
360^\circ2\pi
Why it is true

The definition \displaystyle \theta = \frac{s}{r} only makes sense if the ratio is the same on every circle; otherwise one angle would have many radian measures. It is. The arc length lesson shows, by the similarity of circles, that an n^\circ arc is \displaystyle \frac{n}{360} of the circumference:

\begin{aligned} s &= \frac{n}{360} \cdot 2\pi r \\ \frac{s}{r} &= \frac{n}{360} \cdot 2\pi = \frac{\pi n}{180} \end{aligned}

The radius has canceled. The ratio depends on n alone, so it measures the angle. The same line proves the conversion rule: an angle of n^\circ has radian measure \displaystyle \frac{\pi n}{180}, which is n times \displaystyle \frac{\pi}{180}. Putting n = 360 gives 2\pi and n = 180 gives \pi.

The sector area lesson met the payoff already: with the angle in radians the fraction of the turn is \displaystyle \frac{\theta}{2\pi}, and the sector’s area becomes \displaystyle \frac{1}{2}r^2\theta with no 360 in sight.

Worked examples

Example 1

Minutes and seconds to decimal degrees

The shoulder joint of a robotic arm reports its angle as 47^\circ 21' 36''. Write it in decimal degrees.

Solution

A minute is \displaystyle \frac{1}{60} of a degree and a second is \displaystyle \frac{1}{3600} of a degree, so convert each part to degrees and add:

47 + \frac{21}{60} + \frac{36}{3600} = 47 + 0.35 + 0.01 = 47.36

The angle is 47.36^\circ.

Check. 0.36^\circ is 0.36 \times 60 = 21.6 minutes, and 0.6' is 0.6 \times 60 = 36 seconds, which gives back 21' 36''.

Example 2

Decimal degrees to minutes and seconds

Genowefa’s controller sets the elbow joint to 118.645^\circ. Write it in degrees, minutes and seconds.

Solution

The whole part, 118, is the degrees. The decimal part is a fraction of a degree, and a degree is 60 minutes:

0.645 \times 60 = 38.7 \text{ minutes}

Keep 38 minutes. The leftover 0.7 of a minute is a fraction of a minute, and a minute is 60 seconds:

0.7 \times 60 = 42 \text{ seconds}

So 118.645^\circ = 118^\circ 38' 42''. Going the other way checks it: \displaystyle 118 + \frac{38}{60} + \frac{42}{3600} = 118 + \frac{2322}{3600} = 118.645.

Example 3

Degrees to radians

Write 165^\circ, 252^\circ and -40^\circ in radians, exactly.

Solution

Multiply by \displaystyle \frac{\pi}{180}, which equals 1, and reduce the fraction:

\begin{aligned} 165^\circ &= \frac{165\pi}{180} = \frac{11\pi}{12} \\ 252^\circ &= \frac{252\pi}{180} = \frac{7\pi}{5} \\ -40^\circ &= -\frac{40\pi}{180} = -\frac{2\pi}{9} \end{aligned}

(Divide 165 and 180 by 15, 252 and 180 by 36, and 40 and 180 by 20.) Exact answers keep \pi as a symbol. The sign carries over unchanged: a negative angle is turned the other way, as in angles in standard position, in either unit.

Check. 252^\circ is more than 180^\circ, and \displaystyle \frac{7\pi}{5} is more than \pi, as it should be.

Example 4

Radians to degrees

Write \displaystyle \frac{7\pi}{9} radians and 2.5 radians in degrees. Give the second also in degrees, minutes and seconds, to the nearest second.

Solution

Multiply by \displaystyle \frac{180^\circ}{\pi}. In the first the \pi cancels:

\frac{7\pi}{9} \cdot \frac{180^\circ}{\pi} = 7 \cdot 20^\circ = 140^\circ

The second has no \pi to cancel, so the answer holds \pi in the denominator:

2.5 \cdot \frac{180^\circ}{\pi} = \frac{450^\circ}{\pi} \approx 143.2394^\circ

Now split off minutes and seconds as in Example 2: 0.2394 \times 60 \approx 14.367, so 14', and 0.367 \times 60 \approx 22, so 22''. Carry the unrounded value through, then round only the seconds. The angle is about 143^\circ 14' 22''.

A radian measure with no \pi in it, like 2.5, is just as legitimate as \displaystyle \frac{7\pi}{9}; it is simply an angle that is not a tidy fraction of a turn.

Example 5

Measuring an angle by arc and radius

Hieronim’s robotic arm pivots at a joint O. Its gripper is 0.8 meters from the joint and traces an arc of 1.2 meters as the arm swings. A sensor on the same arm, 0.3 meters from the joint, traces 0.45 meters. Find the angle the arm turned, in radians and in degrees.

\theta
1.2\text{ m}
0.45\text{ m}
0.8\text{ m}
0.3\text{ m}
O
Figure 3Every point on the arm sweeps the same angle, and every point gives the same ratio of arc to radius.
Solution

The arm turns about O, so each point moves on a circle centered at O, and the angle of the swing is a central angle of that circle. By the definition,

\theta = \frac{s}{r} = \frac{1.2}{0.8} = 1.5 \text{ rad}

The sensor gives the same answer, \displaystyle \frac{0.45}{0.3} = 1.5, which is the whole point: every point on the arm turns through one angle, and arc over radius measures that angle no matter which point you use. The meters canceled, so 1.5 has no unit.

In degrees:

1.5 \cdot \frac{180^\circ}{\pi} = \frac{270^\circ}{\pi} \approx 85.94^\circ

just short of a right angle.

Example 6

Comparing angles in mixed units

Put these in order from smallest to largest: 2 radians, 115^\circ, \displaystyle \frac{3\pi}{5} and 114^\circ 35'.

Solution

Angles in different units can only be compared once they share one. Change each to decimal degrees:

\begin{aligned} 2 \cdot \frac{180^\circ}{\pi} &= \frac{360^\circ}{\pi} \approx 114.5916^\circ \\ \frac{3\pi}{5} \cdot \frac{180^\circ}{\pi} &= 108^\circ \\ 114^\circ 35' &= 114 + \tfrac{35}{60} \approx 114.5833^\circ \end{aligned}

The order is \displaystyle \frac{3\pi}{5} < 114^\circ 35' < 2 \text{ rad} < 115^\circ.

Two of them differ by less than a hundredth of a degree: 2 radians is about 114^\circ 35' 30'', just 30'' more than 114^\circ 35'. Rounding 2 radians to 114.6^\circ would have made it look larger than it is and kept the order, but rounding to 115^\circ would have made it equal to the last angle. Round late.

Example 7

A motor that turns in steps

The wrist of Ignacy’s arm is driven by a motor that turns in 200 equal steps per full turn. Find one step in degrees and in radians. The controller asks for a turn of 1.2 radians; how many whole steps come closest, and what angle does the wrist actually turn?

Solution

A full turn is 360^\circ, or 2\pi radians, shared among 200 steps:

\begin{aligned} \frac{360^\circ}{200} &= 1.8^\circ \\ \frac{2\pi}{200} &= \frac{\pi}{100} \approx 0.031416 \text{ rad} \end{aligned}

The number of steps in 1.2 radians is

1.2 \div \frac{\pi}{100} = \frac{120}{\pi} \approx 38.197

The motor can only take whole steps, and 38 is closer than 39. Then the wrist turns

\begin{aligned} 38 \cdot \frac{\pi}{100} &= \frac{19\pi}{50} \approx 1.1938 \text{ rad} \\ 38 \cdot 1.8^\circ &= 68.4^\circ \end{aligned}

The request, 1.2 radians, is about 68.75^\circ, so the wrist falls short by about 0.0062 radians, or 0.35^\circ. The step size, not the arithmetic, limits the accuracy.

Boundaries

The vertex must be at the center. \displaystyle \theta = \frac{s}{r} is about a central angle. An angle whose vertex sits elsewhere, even on the circle, cuts off an arc, but arc over radius is not its measure.

Arc, not chord. The length that counts is along the curve. The straight chord across a 1 radian arc is about 0.959r, shorter than the radius, so a chord over a radius is not the angle.

Same unit on top and bottom. \displaystyle \frac{s}{r} is a plain number only when s and r are measured in the same unit. An arc in centimeters over a radius in meters gives a number 100 times too big.

More than a turn, and turning backward. Radian measure is not stuck between 0 and 2\pi. A point carried twice around a circle travels 4\pi r, an angle of 4\pi, and a clockwise turn is negative in either unit.

Exact and rounded. \displaystyle \frac{\pi}{180} and \displaystyle \frac{180}{\pi} are exact; 57.3 and 0.01745 are not. Keep \pi exact where you can and round once, at the end.

A missing degree sign changes the angle. \theta = 3 means 3 radians, about 172^\circ, not 3^\circ.

Where this goes next

With radians in hand, the arc becomes the radius times the angle, s = r\theta, and a sector’s area \displaystyle \frac{1}{2}r^2\theta; working problems with both is arc length and sector area in radians. Dividing an angle by a time gives angular speed, in radians per second, and the same s = r\theta turns it into the speed of a point on the rim.

On a circle of radius 1 the radian measure of an angle equals the length of its arc. That is the setting of the unit circle, where a turn of \theta radians ends at a point whose coordinates define the cosine and the sine. Placing the angle there starts from standard position, with its vertex at the origin.

Exercises

Degrees, minutes and seconds

  1. Write 23^\circ 45' in decimal degrees.
  2. Write 62^\circ 14' 24'' in decimal degrees.
  3. Write 31.48^\circ in degrees, minutes and seconds.
  4. Write 205.0875^\circ in degrees, minutes and seconds.
  5. Add 38^\circ 47' 50'' and 26^\circ 35' 25''. Give the sum in degrees, minutes and seconds.

Degrees to radians

Write each angle in radians, exactly.

  1. 20^\circ
  2. 105^\circ
  3. 300^\circ
  4. -126^\circ
  5. Write 32^\circ in radians, exactly and then to four decimal places.

Radians to degrees

  1. \displaystyle \frac{5\pi}{12}
  2. \displaystyle \frac{13\pi}{18}
  3. \displaystyle -\frac{9\pi}{4}
  4. 0.75 radians, to two decimal places.
  5. 4 radians, to two decimal places, and then to the nearest second.

Arc over radius

  1. A central angle of a circle of radius 4 cm cuts off an arc of 14 cm. Find the angle in radians and in degrees, to two decimal places.
  2. The gripper of a robotic arm is 0.9 meters from the joint and traces an arc of 0.63 meters. Through what angle did the arm turn, in radians and in degrees to two decimal places?
  3. A circle has radius 5 cm. Find the radian measure of the central angle that cuts off an arc of 5 cm, of 10 cm, and of 2.5 cm.
  4. A joint turns \displaystyle \frac{5}{18} of a full turn. Give the angle in radians and in degrees.
  5. Hieronim rolls a wheel of radius 30 cm along the floor, without slipping, for 1 meter. Through how many radians has the wheel turned? Give the exact value and the angle in degrees to two decimal places.

Comparing and combining

  1. Put in order from smallest to largest: \displaystyle \frac{5\pi}{12}, 1.3 radians, 74^\circ 30'.
  2. Genowefa’s motor turns in 400 equal steps per full turn. Find one step in degrees, in minutes, and in radians. How many whole steps come closest to \displaystyle \frac{\pi}{3}, and what angle in degrees and minutes do they give?

Reasoning

  1. A circle’s radius is measured in centimeters and then in meters. Explain why the radian measure of a central angle comes out the same both times, while the arc length does not.
  2. A central angle of 60^\circ and its two radii make a triangle. Show that its chord is exactly one radius long, and use that to explain why one radian is a little less than 60^\circ.
  3. Ignacy reads a calculator setting of 5 for an angle and says it is a very small angle. What does 5 mean with no degree sign, and how large is it in degrees? Explain why he is wrong.
Answers
  1. 23.75^\circ. \displaystyle 45' = \frac{45}{60} = 0.75 of a degree.
  2. 62.24^\circ. \displaystyle \frac{14}{60} + \frac{24}{3600} = \frac{864}{3600} = 0.24.
  3. 31^\circ 28' 48''. 0.48 \times 60 = 28.8 minutes, and 0.8 \times 60 = 48 seconds.
  4. 205^\circ 5' 15''. 0.0875 \times 60 = 5.25 minutes, and 0.25 \times 60 = 15 seconds.
  5. 65^\circ 23' 15''. Adding part by part gives 64^\circ 82' 75''. Then 75'' = 1' 15'', giving 64^\circ 83' 15'', and 83' = 1^\circ 23'.
  6. \displaystyle \frac{\pi}{9}. \displaystyle \frac{20\pi}{180}, dividing top and bottom by 20.
  7. \displaystyle \frac{7\pi}{12}. \displaystyle \frac{105\pi}{180}, dividing by 15.
  8. \displaystyle \frac{5\pi}{3}. \displaystyle \frac{300\pi}{180}, dividing by 60.
  9. \displaystyle -\frac{7\pi}{10}. \displaystyle -\frac{126\pi}{180}, dividing by 18.
  10. \displaystyle \frac{8\pi}{45} \approx 0.5585. \displaystyle \frac{32\pi}{180}, dividing by 4.
  11. 75^\circ. \displaystyle \frac{5\pi}{12} \cdot \frac{180^\circ}{\pi} = 5 \cdot 15^\circ.
  12. 130^\circ. \displaystyle 13 \cdot \frac{180^\circ}{18} = 13 \cdot 10^\circ.
  13. -405^\circ. \displaystyle -9 \cdot \frac{180^\circ}{4} = -9 \cdot 45^\circ, more than a full turn clockwise.
  14. \displaystyle \frac{135^\circ}{\pi} \approx 42.97^\circ. \displaystyle 0.75 \cdot \frac{180^\circ}{\pi}.
  15. \displaystyle \frac{720^\circ}{\pi} \approx 229.18^\circ \approx 229^\circ 10' 59''. The decimal part 0.18312^\circ is 10.987 minutes, and 0.987' \approx 59''.
  16. 3.5 radians, about 200.54^\circ. \displaystyle \theta = \frac{14}{4}, and \displaystyle 3.5 \cdot \frac{180^\circ}{\pi} = \frac{630^\circ}{\pi}.
  17. 0.7 radians, about 40.11^\circ. \displaystyle \theta = \frac{0.63}{0.9}, and \displaystyle 0.7 \cdot \frac{180^\circ}{\pi} = \frac{126^\circ}{\pi}.
  18. 1, 2 and 0.5 radians. Each is the arc divided by the radius: \displaystyle \frac{5}{5}, \displaystyle \frac{10}{5}, \displaystyle \frac{2.5}{5}.
  19. \displaystyle \frac{5\pi}{9} radians, which is 100^\circ. \displaystyle \frac{5}{18} \cdot 2\pi and \displaystyle \frac{5}{18} \cdot 360^\circ.
  20. \displaystyle \frac{10}{3} \approx 3.33 radians, about 190.99^\circ. Rolling without slipping, each point of the rim touches the floor once, so the arc that rolled past is 100 cm. Then \displaystyle \theta = \frac{100}{30} (both in centimeters), and \displaystyle \frac{10}{3} \cdot \frac{180^\circ}{\pi} = \frac{600^\circ}{\pi}.
  21. 1.3 radians \displaystyle < 74^\circ 30' < \frac{5\pi}{12}. In degrees they are \displaystyle \frac{234^\circ}{\pi} \approx 74.48^\circ, 74.5^\circ, and 75^\circ.
  22. One step is 0.9^\circ = 54', or \displaystyle \frac{\pi}{200} radians. \displaystyle \frac{\pi}{3} \div \frac{\pi}{200} = \frac{200}{3} \approx 66.67, so 67 steps, giving 67 \times 0.9^\circ = 60.3^\circ = 60^\circ 18', which is 18' more than the 60^\circ asked for.
  23. Changing the unit multiplies both s and r by 100 (meters to centimeters), so the arc length changes, but the ratio \displaystyle \frac{s}{r} does not: the 100 cancels. A radian measure is a ratio of two lengths, so it carries no unit and does not depend on the unit chosen.
  24. The triangle has two sides of length r, so its base angles are equal, and they share 180^\circ - 60^\circ = 120^\circ, so each is 60^\circ. All three angles are 60^\circ, the triangle is equilateral, and the chord is r. The arc joins the same two points along a curve, so it is longer than the straight chord: s > r. Then the radian measure of 60^\circ is \displaystyle \frac{s}{r} > 1, so one radian is a smaller angle than 60^\circ. (In fact 1 \text{ rad} \approx 57.30^\circ.)
  25. An angle with no degree sign is in radians, so 5 means 5 radians: \displaystyle 5 \cdot \frac{180^\circ}{\pi} = \frac{900^\circ}{\pi} \approx 286.48^\circ, more than three quarters of a full turn. A radian is about 57^\circ, so a radian measure is a far bigger angle than the same number of degrees.
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