Trigonometric FunctionsAngles

An angle measures a rotation. Pin one end of a ray and swing the ray round, like the hand of a clock. Where the ray starts is the initial side of the angle, where it stops is the terminal side, and the fixedmovingmiddle point it turns about is the vertex. The direction of the turn matters: an anticlockwise turn makes a positive angle, and a clockwise turn makes a negative angle.

Fig. 3.1 — OA is the initial side and OB the terminal side in both pictures.

The size of an angle is simply how much the ray turned to carry the terminal side away from the initial side. A length can be measured in centimetres or in inches; in the same way an angle can be measured in several units. The definition itself suggests the first one: one complete revolution, in which the ray swings all the way round and lands back on the initial side.

Turns of Any Size

Whole revolutions are handy for big turns — a ceiling fan might make 8 revolutions every second — but most angles are only part of a turn, and some are more than one.

Fig. 3.2 and Fig. 3.3, live. Drag B, the end of the terminal side. Turn it anticlockwise for positive angles and clockwise for negative ones, and keep going past a full turn: the spiral counts every revolution. Complete all four missions.

420° and 60° end on exactly the same terminal side, yet they are different angles: one of them made an extra revolution on the way. For everyday angles we need a unit finer than a whole revolution. The two units used most are the degree and the radian.

Degree measure

The degree slices one complete revolution into 360 equal parts. A rotation of of a revolution is one degree, written 1°. For finer work a degree is split into 60 equal minutes (1′), and a minute into 60 equal seconds (1″) — the same sixty-way split as on a clock, borrowed for angles.

Quick check: half a degree is 30505 minutes, so 40.5° = 40° 30′. A quarter of a minute is 15254 seconds.

The angles you made in Fig. 3.3 — 360°, −30°, 420° and −420° — were all measured in degrees, and so are the right angle (90°) and the straight angle (180°).

Radian measure

The second unit, the radian, is the one mathematicians prefer, because it ties an angle directly to the arc it cuts off. Swing a radius OA of a circle round to OB. If the arc AB is exactly as long as the radius, the angle AOB is 1 radian. On a unit circle — a circle of radius 1 unit — this says: the angle that an arc of length 1 subtends at the centre is 1 radian.

Fig. 3.4, live. Drag B round the circle. The purple arc is measured in radius-lengths: the numbered notches mark 1, 2, 3, … radii of arc. Complete all three missions.

The circumferencediameterarea of a unit circle is , so one complete revolution is radians and half a turn is radians — just over three radius-lengths, as you saw.

In a circle of radius , an arc of length subtends 1 radian. Equal arcs subtend equal angles, so an arc of length subtends radians.

So an arc of 15 cm on a circle of radius 6 cm subtends 2.5900.4 radians. A radian measure is a length divided by a length, so it is a pure number with no units — one reason it slots so neatly into calculus and physics.

Relation between radian and real numbers

Take the unit circle with centre O and a point A on it, and use OA as the initial side. Because the radius is 1, the length of any arc measured from A is exactly the radian measure of the angle it subtends at O. Now draw the line that touches the circle at A. Let A stand for the real number 0, let the part of the line above A carry the positive numbers and the part below A carry the negative numbers (Fig. 3.5).

Fig. 3.5, live. Wind the upper part anticlockwise and the lower part clockwise round the circle, like thread round a spool.

So the real number π lands exactly where the angle of π radians — half a turn — ends. Radian measures and real numbers are two names for the same thing, and this is what will let us feed any real number into sine and cosine.

Relation between degree and radian

One complete revolution has two names: 360° and radians. Setting them equal builds the bridge between the two units:

With :

So one radian is a little less than 60°45°90° — just what the purple arc showed you. Here are the radian measures of some common angles:

Try it: 225° = 5π/44π/55π/2 radian, and radian = 300°150°330°.

Converting Both Ways

Example 1 : Convert 40° 20′ into radian measure.

Solution : First turn the minutes into a fraction of a degree: 20′ = degree = 1/31/22/3 degree. So 40° 20′ = degree = degree. Now use 180° = π radian:

121π/540121π/18040π/180 radian

Example 2 : Convert 6 radians into degree measure.

Solution : π radian = 180°, so

The whole part gives 343°. The fraction of a degree is minutes, i.e. 38427 minutes and of a minute. And of a minute is seconds, which rounds to 11102 seconds.

So 6 radians ≈ 343° 38′ 11″ — almost a full turn, which makes sense: a full turn is radians.

Arcs at Work

Example 3 : Find the radius of the circle in which a central angle of 60° intercepts an arc of length 37.4 cm (use ).

Solution : Here cm and π/3π/6π/2 radian. From we get :

35.737.426.2 cm

Example 4 : The minute hand of a watch is 1.5 cm long. How far does its tip move in 40 minutes? (Use .)

Solution : The minute hand makes one full revolution in 60 minutes, so in 40 minutes it turns through 2/31/33/4 of a revolution:

So the tip moves

6.283.144.71 cm

Example 5 : If arcs of the same length in two circles subtend angles of 65° and 110° at their centres, find the ratio of the radii.

Solution : Let the radii be and . In radians,

The arcs are equal, so :

Hence 22 : 1313 : 2265 : 110.

Speed Round: Degrees and Radians

Convert in your head: multiply by one way and by the other. After each answer you will see the working.

Get five right to move on.