Fundamentals of Trigonometry

Lessons

Senior 4 · Mathematics

Fundamentals of Trigonometry

5 lessons

Lesson 1 of 5

Angles and their measurement

25 min

How far does a wheel roll?

A bicycle taxi's back wheel has a radius of about 33 cm. When the wheel turns through some angle, the bicycle moves forward by the length of the arc that touches the road. If the angle is measured in degrees, finding that distance needs a conversion every time. Measure the angle instead by "how many radii of arc" it cuts off, and the distance is simply radius × angle. That second unit, the radian, is the one mathematicians and engineers use, and it is the unit in which the rest of your A-level trigonometry and calculus is written.

Definition

Oriented angle: an angle formed by rotating a ray, the initial side, about its endpoint (the vertex) to a final position, the terminal side. The angle is positive for an anticlockwise rotation and negative for a clockwise one. It is in standard position when the vertex is at the origin and the initial side lies along the positive $x$ -axis.

Coterminal angles: angles in standard position with the same terminal side; they differ by a whole number of turns, $360^\circ k$ or $2\pi k$ , $k \in \mathbb{Z}$ .

Definition

Units of angle

Degree: $\frac{1}{360}$ of a full turn, written $1^\circ$ . It is divided into 60 minutes ( $1^\circ = 60'$ ) and each minute into 60 seconds ( $1^{'} = 60^{''}$ ).
Radian: the angle at the centre of a circle subtended by an arc whose length equals the radius. A full turn is the whole circumference, $2\pi r$ , which is $2\pi$ radii, so a full turn is $2\pi$ rad.
Grade (gon): $\frac{1}{400}$ of a full turn, used by some surveyors; a right angle is 100 grades.

360^\circ = 2\pi \text{ rad} = 400 \text{ gr}, \qquad \pi \text{ rad} = 180^\circ, \qquad 1 \text{ rad} = \frac{180^\circ}{\pi} \approx 57.3^\circ .

One radian

To convert, multiply degrees by $\frac{\pi}{180}$ to get radians, and radians by $\frac{180}{\pi}$ to get degrees. Learn the common angles; they appear in every later lesson.

Degrees	0°	30°	45°	60°	90°	120°	135°	150°	180°	270°	360°
Radians	0	$\frac{\pi}{6}$	$\frac{\pi}{4}$	$\frac{\pi}{3}$	$\frac{\pi}{2}$	$\frac{2\pi}{3}$	$\frac{3\pi}{4}$	$\frac{5\pi}{6}$	$π \pi$	$\frac{3\pi}{2}$	$2\pi$

Worked example 1: converting

(a) Convert $135^\circ$ and $47^\circ 30'$ to radians. (b) Convert $\frac{5\pi}{12}$ and $2.5$ rad to degrees, giving the second in degrees and minutes.

Solution

$135 \times \frac{\pi}{180} = \frac{3\pi}{4}$ rad. $47^\circ 30' = 47.5^\circ$ , and $47.5 \times \frac{\pi}{180} \approx 0.8290$ rad.
$\frac{5\pi}{12} \times \frac{180}{\pi} = 75^\circ$ . $2.5 \times \frac{180}{\pi} \approx 143.24^\circ$ ; since $0.24^\circ \times 60 \approx 14'$ , this is about $143^\circ 14'$ .

Arc length and sector area

In a circle of radius $r$ , the arc and the sector cut off by a central angle $\theta$ are the fraction $\frac{\theta}{2\pi}$ of the whole circumference and of the whole disc. With $\theta$ in radians:

s = \frac{\theta}{2\pi} \times 2\pi r = r\theta, \qquad A = \frac{\theta}{2\pi} \times \pi r^2 = \frac{1}{2} r^2 \theta .

These short formulas are the reason radians are used: in degrees the same facts read $s = \frac{\pi r \theta}{180}$ and $A = \frac{\pi r^2 \theta}{360}$ .

Worked example 2: the bicycle wheel

A wheel of radius $0.33\,\mathrm{m}$ turns through $1500^\circ$ . (a) How far does the bicycle move? (b) How many complete turns does the wheel make in 1 km?

Solution

$\theta = 1500 \times \frac{\pi}{180} = \frac{25\pi}{3} \approx 26.18$ rad, so $s = r\theta = 0.33 \times 26.18 \approx 8.64\,\mathrm{m}$ .
One turn covers $2\pi r = 0.66\pi \approx 2.073\,\mathrm{m}$ , so 1 km needs $\frac{1000}{2.073} \approx 482.3$ turns: 482 complete turns.

Worked example 3: a sector

A garden sprinkler waters the ground up to $6\,\mathrm{m}$ away while it sweeps through $120^\circ$ . Find the area watered and the length of the curved edge.

Solution $\theta = \frac{2\pi}{3}$ . Area $A = \frac{1}{2} \times 6^2 \times \frac{2\pi}{3} = 12\pi \approx 37.7\,\mathrm{m^2}$ . Curved edge $s = 6 \times \frac{2\pi}{3} = 4\pi \approx 12.6\,\mathrm{m}$ .

Worked example 4: principal measure

The principal measure of an angle is its coterminal angle in $]-\pi, \pi]$ (or $]-180^\circ, 180^\circ]$ ). Find it for (a) $\frac{17\pi}{4}$ (b) $-1000^\circ$ .

Solution Add or subtract whole turns.

$\frac{17\pi}{4} - 2 \times 2\pi = \frac{17\pi}{4} - \frac{16\pi}{4} = \frac{\pi}{4}$ .
$-1000^\circ + 3 \times 360^\circ = 80^\circ$ .

Warning

Check your calculator's mode. $\sin 30$ is 0.5 in DEG mode but about $- 0.988$ in RAD mode, because there it means the sine of 30 radians. Use radians in $s = r\theta$ and $A = \frac{1}{2}r^2\theta$ ; with degrees these formulas give nonsense.

Tip

Remember one fact, $180^\circ = \pi$ rad, and every conversion follows: $30^\circ$ is a sixth of $180^\circ$ , so it is $\frac{\pi}{6}$ ; $\frac{3\pi}{4}$ is three quarters of $180^\circ$ , so it is $135^\circ$ .

NancyPi: Radians and Degrees. Watch for what a radian measures and for the two conversion factors, π/180 and 180/π.

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Practice

Give exact answers where you can; otherwise 3 significant figures.

Convert to radians: (a) $210^\circ$ (b) $315^\circ$ (c) $-150^\circ$ .
Convert to degrees: (a) $\frac{5\pi}{12}$ (b) $1.2$ rad (in degrees and minutes).
(a) Write $38.4^\circ$ in degrees and minutes. (b) Write $72^\circ 15' 36''$ as a decimal number of degrees.
Find the principal measure of (a) $-\frac{11\pi}{3}$ (b) $1000^\circ$ .
A sector has radius 12 cm and angle 2.4 rad. Find its arc length and area.
The minute hand of a clock is 15 cm long. How far does its tip travel in 20 minutes, and what area does the hand sweep?
A sector of radius 10 cm has area $50\,\mathrm{cm^2}$ . Find its angle in radians and in degrees.
A bicycle wheel of radius 0.3 m turns at 300 revolutions per minute. Find its angular speed in rad/s and the speed of the bicycle in m/s.

Show answersHide answers

(a) $\frac{7\pi}{6}$ (b) $\frac{7\pi}{4}$ (c) $-\frac{5\pi}{6}$ .
(a) $75^\circ$ (b) $68.75^\circ \approx 68^\circ 45'$ .
(a) $0.4^\circ = 24'$ , so $38^\circ 24'$ . (b) $72 + \frac{15}{60} + \frac{36}{3600} = 72.26^\circ$ .
(a) $-\frac{11\pi}{3} + 4\pi = \frac{\pi}{3}$ (b) $1000^\circ - 3 \times 360^\circ = -80^\circ$ .
$s = 12 \times 2.4 = 28.8\,\mathrm{cm}$ ; $A = \frac{1}{2} \times 144 \times 2.4 = 172.8\,\mathrm{cm^2}$ .
20 minutes is a third of a turn, $\frac{2\pi}{3}$ : $s = 10\pi \approx 31.4\,\mathrm{cm}$ ; $A = 75\pi \approx 236\,\mathrm{cm^2}$ .
$\theta = \frac{2A}{r^2} = \frac{100}{100} = 1$ rad $\approx 57.3^\circ$ .
$\omega = \frac{300 \times 2\pi}{60} = 10\pi \approx 31.4$ rad/s; $v = r\omega = 3\pi \approx 9.42\,\mathrm{m/s}$ .

Quick check

Which of these is 135° in radians?

Key points

$π \pi$ rad $= 180^\circ = 200$ gr; $1^\circ = 60'$ , $1^{'} = 60^{''}$ .
Anticlockwise angles are positive; coterminal angles differ by $2k\pi$ .
With $\theta$ in radians: arc $s = r\theta$ , sector area $A = \frac{1}{2}r^2\theta$ .

Sources and credits

Written by Ganzaa: the explanations, the diagrams, the worked examples (the wheel and sprinkler sizes are illustrative), the practice items and the questions. The unit follows the topic list of REB's S4 Core Mathematics (2026–2027); no REB text is reproduced.
Photo: Taxi bicycle, Alex shema, Wikimedia Commons, CC BY-SA 4.0.
Video: Radians and Degrees, NancyPi, embedded from YouTube.
This lesson adapts CC BY-SA material and is shared under CC BY-SA 4.0.

1How far does a wheel roll?

2One radian

3Arc length and sector area

How far does a wheel roll?

One radian

Arc length and sector area