Unit 6, lesson 44

Radians and Angle Measurement

What a radian measures, how to move between radians and degrees, how to find arc length on circles of any size, and how signs and extra full turns describe a rotation.

  • Theory part 87
  • Practice part 88
  • Angles and the Unit Circle

Take a piece of string as long as the radius of a circle and lay it along the edge. The angle it covers at the center is one radian, and it is the same angle on a coin or on a Ferris wheel. Measuring angles this way ties them directly to lengths along the circle, which is why calculus uses radians almost everywhere. Looking ahead, calculus proves that sin⁡xx\frac{\sin x}{x} approaches 11 as xx approaches 00 and that the slope of sin⁡x\sin x is cos⁡x\cos x. Both facts hold in this simple form only when xx is measured in radians.

Key terms

Angle

Definition An angle is formed by rotating a ray about its endpoint, called the vertex, from an initial side to a terminal side. An angle is in standard position when its vertex is at the origin and its initial side lies along the positive xx-axis. The measure of the angle records how much rotation took place and in which direction.

In plain words An angle is an amount of turning. You start along the positive xx-axis and rotate to wherever the terminal side ends up.

Example A quarter turn from the positive xx-axis puts the terminal side on the positive yy-axis. A half turn puts it on the negative xx-axis. Two different rotations can end at the same terminal side, which is why the amount of turning, not only the final position, is part of the angle.

Radius and arc length

Definition A circle of radius r>0r > 0 is the set of points at distance rr from its center. When a central angle is drawn, its sides cut off an arc of the circle, and the arc length ss is the distance measured along the circle from one end of the arc to the other.

In plain words The radius is the distance from the center to the edge. Arc length is how far an ant would walk along the edge, not the straight line distance across.

Example A circle of radius 55 has circumference 2π⋅5=10π2\pi \cdot 5 = 10\pi. The arc cut off by a quarter turn is one quarter of that, so its length is

s=10π4=5π2≈7.85.s = \frac{10\pi}{4} = \frac{5\pi}{2} \approx 7.85.

The straight chord between the same two points is only 52≈7.075\sqrt{2} \approx 7.07.

Radian

Definition Place the vertex of an angle at the center of a circle of radius r>0r > 0. As the ray rotates counterclockwise from the initial side to the terminal side, the point where it meets the circle travels a distance s>0s > 0 along the circle, counting every full lap. The radian measure of this positive angle is

θ=sr.\theta = \frac{s}{r}.

The ratio does not depend on the size of the circle, because scaling the circle scales ss and rr by the same factor. One radian is the angle whose arc is exactly as long as the radius. Since ss and rr are both lengths, a radian measure is a pure number. Solving for ss gives the arc length formula s=rθs = r\theta for a positive angle in radians.

In plain words The radian measure counts how many radii fit along the arc.

Example On a circle of radius 22, an arc of length 33 gives θ=32=1.5\theta = \frac{3}{2} = 1.5 radians. Scale the same wedge up to a circle of radius 66. The arc becomes 99 and the angle is still 96=1.5\frac{9}{6} = 1.5 radians. On a circle of radius 1010 the same wedge has arc length 10⋅1.5=1510 \cdot 1.5 = 15.

Full turn

Definition One complete counterclockwise turn sweeps out the whole circumference 2πr2\pi r, so its radian measure is

2πrr=2π.\frac{2\pi r}{r} = 2\pi.

In degrees a full turn is 360∘360^\circ. A half turn is π\pi radians and a quarter turn is π2\frac{\pi}{2} radians.

In plain words Going once around any circle is about 6.286.28 radii of walking, so a full turn is 2π2\pi radians.

Example A wheel turns 33 full times. In radians that is 3⋅2π=6π≈18.853 \cdot 2\pi = 6\pi \approx 18.85. A third of a turn is 2π3\frac{2\pi}{3} radians.

Degree

Definition A degree is the angle unit equal to 1360\frac{1}{360} of a full turn, written 1∘1^\circ. Degree measure is a convention chosen by dividing the circle into 360360 equal parts, while radian measure comes from the circle itself as a ratio of lengths.

In plain words Degrees chop a full turn into 360360 equal slices. They are handy for everyday angles but have no built-in link to arc length.

Example A right angle is a quarter turn, which is 3604=90∘\frac{360}{4} = 90^\circ. An equilateral triangle has three equal angles adding up to 180∘180^\circ, so each is 60∘60^\circ.

Angle conversion

Definition Because a full turn is both 2π2\pi radians and 360∘360^\circ, we have π radians=180∘\pi \text{ radians} = 180^\circ. To convert,

radians=degrees⋅π180,degrees=radians⋅180π.\begin{gathered} \text{radians} = \text{degrees} \cdot \frac{\pi}{180}, \\[0.4em] \text{degrees} = \text{radians} \cdot \frac{180}{\pi}. \end{gathered}

In plain words Multiply by a fraction equal to one, with the unit you want on top and the unit you have on the bottom.

Example Turning 150∘150^\circ into radians and π12\frac{\pi}{12} into degrees,

150⋅π180=5π6,π12⋅180π=15∘.\begin{gathered} 150 \cdot \frac{\pi}{180} = \frac{5\pi}{6}, \\[0.4em] \frac{\pi}{12} \cdot \frac{180}{\pi} = 15^\circ. \end{gathered}

One radian is 180π≈57.30∘\frac{180}{\pi} \approx 57.30^\circ, a little less than a sixth of a turn.

Signed rotation, counterclockwise and clockwise

Definition An angle measure carries a sign that records the direction of rotation. A counterclockwise rotation from the initial side has positive measure, and a clockwise rotation has negative measure. For a rotation through a signed angle θ\theta in radians on a circle of radius r>0r > 0, the length of the path traced along the circle is

s=r∣θ∣.s = r \lvert \theta \rvert.

The signed quantity rθr\theta is also useful as an arc coordinate, positive for counterclockwise travel and negative for clockwise travel.

In plain words Counterclockwise is the positive direction and clockwise is the negative one. The sign tells you which way you turned, and the absolute value tells you how far.

Example The angle −3π4-\frac{3\pi}{4} is a clockwise rotation of 135∘135^\circ, ending in the third quadrant. On a circle of radius 44 the point travels a distance of

4⋅∣−3π4∣=3π≈9.42,4 \cdot \left\lvert -\frac{3\pi}{4} \right\rvert = 3\pi \approx 9.42,

and its signed arc coordinate is −3π-3\pi.

Coterminal angles

Definition Two angles in standard position are coterminal when they have the same terminal side. In radians, α\alpha and β\beta are coterminal exactly when

β=α+2πkfor some integer k.\beta = \alpha + 2\pi k \quad \text{for some integer } k.

In degrees the condition is β=α+360∘k\beta = \alpha + 360^\circ k. Coterminal angles describe the same final position but different amounts of rotation.

In plain words Adding or removing whole turns does not change where you end up, only how you got there.

Example Consider the angle −7π3-\frac{7\pi}{3}. It equals −π3−2π-\frac{\pi}{3} - 2\pi, a clockwise sixth of a turn followed by one extra full clockwise turn. Adding 4π4\pi gives the coterminal angle 5π3\frac{5\pi}{3} in [0,2π)[0, 2\pi). On a circle of radius 33 the rotation −7π3-\frac{7\pi}{3} traces a path of length

3⋅7π3=7π≈21.99,3 \cdot \frac{7\pi}{3} = 7\pi \approx 21.99,

while the rotation 5π3\frac{5\pi}{3} traces only 5π≈15.715\pi \approx 15.71. Same terminal side, different distance traveled.

Check yourself

  1. A wheel of radius 0.30.3 meters turns about its center so that a point on its rim travels 0.750.75 meters along the circle. Through what angle did the wheel turn, in radians and in degrees?
Show the answer

The angle is

θ=0.750.3=2.5 radians,\theta = \frac{0.75}{0.3} = 2.5 \text{ radians,}

which is 2.5⋅180π≈143.2∘2.5 \cdot \frac{180}{\pi} \approx 143.2^\circ.

  1. Convert 225∘225^\circ to radians and 7π6\frac{7\pi}{6} to degrees.
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The conversions are 225⋅π180=5π4225 \cdot \frac{\pi}{180} = \frac{5\pi}{4} and 7π6⋅180π=210∘\frac{7\pi}{6} \cdot \frac{180}{\pi} = 210^\circ.

  1. Describe the rotation −11π4-\frac{11\pi}{4}, find the coterminal angle in [0,2π)[0, 2\pi), and find the distance traveled along a circle of radius 22.
Show the answer

Since −11π4=−2π−3π4-\frac{11\pi}{4} = -2\pi - \frac{3\pi}{4}, it is one full clockwise turn followed by a further 135∘135^\circ clockwise. Adding 4π4\pi gives the coterminal angle 5π4\frac{5\pi}{4}. The distance traveled is

2⋅11π4=11π2≈17.28.2 \cdot \frac{11\pi}{4} = \frac{11\pi}{2} \approx 17.28.

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