Why Mathematicians Measure Angles in Radians Instead of Degrees

In earlier classes, sine and cosine were ratios of sides in a right-angled triangle, so angles stopped at $90°$. Wheels, clock hands and waves turn through far larger angles, and even negative ones. Defining trigonometric functions on a circle extends them to every angle.

This part covers radian measure and arc length, unit circle definitions, sector area, and signs in the four quadrants.

**One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius, $\pi$ radians equal $180°$, and an arc subtending $\theta$ radians in a circle of radius r has length $s=r\theta$.

Conversions:**

$$\text{radians}=\text{degrees}\times \frac{\pi }{180},\text{degrees}=\text{radians}\times \frac{180}{\pi }$$

- $1\text{ rad}\approx 57.3°$
- $30°=\frac{\pi }{6}$, $45°=\frac{\pi }{4}$, $60°=\frac{\pi }{3}$, $90°=\frac{\pi }{2}$

Worked example. Convert $135°$ to radians: $135\times \frac{\pi }{180}=\frac{3\pi }{4}$ rad. Convert $2.5$ rad to degrees: $2.5\times \frac{180}{\pi }\approx 143.2°$.

Worked example 2. A pendulum 42 cm long swings through $36°$. Since $36°=\frac{\pi }{5}$ rad, the arc traced by its bob is

$$s=r\theta =42\times \frac{\pi }{5}=42\times \frac{22}{35}=26.4\text{ cm}$$

An everyday example. The minute hand of a wall clock, 10 cm long, turns through $\frac{\pi }{3}$ rad in 10 minutes, so its tip travels $10\times \frac{\pi }{3}\approx 10.5$ cm.

The substance. **$s=r\theta$ works only with $\theta$ in radians** — substituting 36 instead of $\frac{\pi }{5}$ gives an absurdly long arc.

**If the terminal side of an angle $\theta$ meets the unit circle $x^2+y^2=1$ at $P(x,y)$, then $\cos \theta =x$ and $\sin \theta =y$, the other four functions follow as ratios, and because P lies on the circle, ${\sin }^2\theta +{\cos }^2\theta =1$ for every angle.

Definitions** for $P(x,y)$ on the unit circle:

- $\sin \theta =y$ and $\cos \theta =x$
- $\tan \theta =\frac{y}{x}$ for $x\ne 0$, and $\cot \theta =\frac{x}{y}$ for $y\ne 0$
- $\sec \theta =\frac{1}{x}$ for $x\ne 0$, and $\text{cosec}\theta =\frac{1}{y}$ for $y\ne 0$

Proof of the identity. P lies on the unit circle, so $x^2+y^2=1$. Substituting $x=\cos \theta$ and $y=\sin \theta$ gives

$${\sin }^2\theta +{\cos }^2\theta =1$$

for every angle, including obtuse and negative angles. Dividing by ${\cos }^2\theta$ gives $1+{\tan }^2\theta ={\sec }^2\theta$.

Worked example. If $\sin \theta =\frac{3}{5}$ and $\theta$ lies in the second quadrant, then ${\cos }^2\theta =1-\frac{9}{25}=\frac{16}{25}$, so $\cos \theta =-\frac{4}{5}$ and $\tan \theta =-\frac{3}{4}$.

An everyday example. A seat on a giant wheel at a mela traces a circle; its height and sideways position at each moment are the sine and cosine of the angle turned.

The substance. **Right-triangle definitions stop at $90°$, the unit circle does not** — at $90°$ the point is $(0,1)$, so $\sin 90°=1$ and $\cos 90°=0$.

**A sector with central angle $\theta$ radians in a circle of radius r has area $A=\frac{1}{2}{r}^2\theta$, which can also be written as $A=\frac{1}{2}rs$ using the arc length s.

Why it works.** A full circle has angle $2\pi$ and area $\pi r^2$. A sector takes the fraction $\frac{\theta }{2\pi }$ of the circle:

$$A=\frac{\theta }{2\pi }\times \pi r^2=\frac{1}{2}{r}^2\theta$$

Worked example. Find the area of a sector of radius 14 cm and angle $\frac{\pi }{4}$:

$$A=\frac{1}{2}\times 14^2\times \frac{\pi }{4}=24.5\pi \approx 77{\text{ cm}}^2$$

Worked example 2. A sector of radius 12 cm has perimeter 40 cm. Its arc is $40-24=16$ cm, so $\theta =\frac{16}{12}=\frac{4}{3}$ rad, and its area is $\frac{1}{2}\times 12\times 16=96{\text{ cm}}^2$.

An everyday example. **A garden sprinkler sweeping $120°$ with a 3 m jet** waters $\frac{1}{2}\times 3^2\times \frac{2\pi }{3}=3\pi \approx 9.4{\text{ m}}^2$ of lawn.

The substance. A sector's perimeter includes both radii — it is $2r+s$, not just the arc.

**All six functions are positive in the first quadrant, only sine and cosecant in the second, only tangent and cotangent in the third, and only cosine and secant in the fourth; sine and cosine have range $[-1,1]$, while tangent takes every real value.

Signs by quadrant:

- First — all positive
- Second — sin and cosec positive
- Third — tan and cot positive
- Fourth — cos and sec positive

Domains and ranges:**

- $\sin x$ and $\cos x$ — domain $\mathbb{R}$, range $[-1,1]$
- $\tan x$ — domain excludes odd multiples of $\frac{\pi }{2}$; range $\mathbb{R}$
- $\sec x$ — same domain as $\tan x$; range $(-\infty ,-1]\cup [1,\infty )$
- $\cot x$ and $\text{cosec}x$ — domain excludes multiples of $\pi$

Worked example. Find $\sin 240°$. Since $240°=180°+60°$ lies in the third quadrant, sine is negative:

$$\sin 240°=-\sin 60°=-\frac{\sqrt{3}}{2}\approx -0.866$$

An everyday example. The tip of a ceiling fan blade is above the centre in the first two quadrants and below it in the last two — exactly the sign pattern of sine.

The substance. **$\sin x=2$ has no solution** — the range of sine is $[-1,1]$ for every angle.