Circular System (Radian Measure)

Circular System (Radian Measure)

In the circular system, the unit of angle measurement is the radian. Let a circle have centre $O$ and radius $r$, and let $A$, $B$ be two points on the circle such that the length of arc $AB$ is exactly equal to $r$. The central angle $\angle AOB$ subtended by such an arc is defined to measure one radian, written $1^{c}$.

One radian is the measure of the angle subtended at the centre of a circle by an arc whose length equals the radius of the circle.

The radian is independent of the radius, and $\pi^{c} = 180°$

Theorem. The radian, as defined above, does not depend on which circle (i.e. which radius) is used to define it, and $\pi^{c} = 180°$.

Proof (in outline). Take a circle of centre $O$ and radius $r$, and let $AB$ be an arc of length $r$, so $m\angle AOB = 1^{c}$ by definition. Produce $AO$ beyond $O$ to meet the circle again at $C$; then $AC$ is a diameter, and $\angle AOC$ is a straight angle, i.e. $\angle AOC = 2$ right angles $= 180°$.

At the centre of a circle, angles are proportional to the arcs that subtend them, so

(since arc $ABC$ is a semicircle of length $\pi r$). Hence

which is a constant that does not involve $r$ at all -- so one radian is a well-defined, radius-independent unit, and

Converting between degree and radian measure

If the same angle has measure $r$ in radian and $\theta$ in degree, then, because both are measured as a proportion of the same straight angle,

• Degree $\to$ radian: multiply the degree measure by $\dfrac{\pi}{180}$.
• Radian $\to$ degree: multiply the radian measure by $\dfrac{180}{\pi}$.

Taking $\pi \approx 3.14$, one radian works out to

and the fractional part of the degree can be turned into minutes and seconds in the usual way: $0.3248° = (0.3248\times 60)' = 19.488' = 19' + (0.488\times 60)'' \approx 19'\,29''$, so $1^{c} \approx 57°\,19'\,29''$.

A quick-reference table of some common angles in both systems:

Application: the angle turned by the hands of a clock

Both hands of a clock complete one full rotation ($360°$) but at different fixed rates: the minute hand in $60$ minutes, the hour hand in $12$ hours. This gives their rotation rates:

Caution: the word "minute" is used both for a unit of time and for $\frac{1}{60}$ of a degree -- these are two completely different things, and the context (clock reading vs. angle measure) always tells them apart.

Solved Examples

Example 1 (degree $\to$ radian). Convert (i) $70°$ (ii) $-120°$ (iii) $\frac{1}{4}°$ to radian.

Using $\theta° = \theta\times\frac{\pi}{180}$ radian:

1. $70° = 70\times\frac{\pi}{180} = \frac{7\pi}{18}$ radian.
2. $-120° = -120\times\frac{\pi}{180} = -\frac{2\pi}{3}$ radian.
3. $\frac{1}{4}° = \frac{1}{4}\times\frac{\pi}{180} = \frac{\pi}{720}$ radian. Example 2 (radian $\to$ degree). Convert (i) $\frac{7\pi}{3}$ (ii) $-\frac{\pi}{18}$ (iii) $\frac{4}{7}$ radian to degree. Using $\theta^{c} = \theta\times\frac{180}{\pi}$ degree:

(i) $\frac{7\pi}{3}\times\frac{180}{\pi} = 7\times 60 = 420°$.

(ii) $-\frac{\pi}{18}\times\frac{180}{\pi} = -10°$.

(iii) $\frac{4}{7}\times\frac{180}{\pi} = \frac{720}{7\pi}° \approx \frac{360}{11}°$ (using $\pi\approx\frac{22}{7}$).

Example 3 (decimal degree $\to$ D-M-S). Express (i) $74.87°$ (ii) $-30.6947°$ in degree-minute-second form.

(i) $74.87° = 74° + 0.87° = 74° + (0.87\times 60)' = 74° + 52.2' = 74°52' + (0.2\times 60)'' = 74°52'12''$.

(ii) $-30.6947° = -[30° + 0.6947°] = -[30° + 41.682'] = -[30°41' + 0.682\times 60''] = -[30°41'40.92''] \approx -30°41'41''$.

Example 4 (triangle angles in A.P.). The angles of a triangle are in A.P. and the smallest is $40°$; find all three angles in degree and radian.

Let the angles be $a-d, a, a+d$. Angle sum: $3a = 180° \Rightarrow a = 60°$. Smallest angle: $a-d = 40° \Rightarrow d = 20°$. So the angles are $40°, 60°, 80°$, i.e. in radian $\frac{2\pi}{9}, \frac{\pi}{3}, \frac{4\pi}{9}$.

Example 5 (difference of acute angles in a right triangle). The difference of the two acute angles of a right triangle is $\frac{7\pi}{30}$ radian; find the angles in degree.

$\frac{7\pi}{30}$ radian $= \frac{7\pi}{30}\times\frac{180}{\pi} = 42°$, so with the acute angles $x,y$: $x-y=42°$ and (right triangle) $x+y=90°$. Adding, $2x = 132° \Rightarrow x = 66°$, then $y = 24°$. Angles of the triangle: $66°, 90°, 24°$.

Example 6 (quadrilateral, one angle in radian, rest in ratio). One angle of a quadrilateral is $\frac{2\pi}{9}$ radian, the other three are in ratio $3:5:8$; find all in degree. …