Angles of any size

What you will be able to do

• describe an angle as a turn, with an initial side and a terminal side;
• work with negative angles and with angles of more than one revolution;
• find the angle from $0^\circ$ to $360^\circ$ that ends in the same place, and its quadrant;
• change between decimal degrees and degrees, minutes and seconds.

An angle is a turn

In Class 10, an angle of a right triangle always lay between $0^\circ$ and $90^\circ$. A wheel, a fan or the hand of a clock can turn much further, and in either direction. So we now think of an angle as an amount of rotation.

A ray starts along the initial side and turns about its end point, the vertex. Where it stops is the terminal side. The angle measures how far it turned.

• An anticlockwise turn gives a positive angle.
• A clockwise turn gives a negative angle.

We usually put the vertex at the origin, with the initial side along the positive $x$-axis. The angle is then in standard position.

More than one turn

One complete turn, back to the start, is one revolution, $360^\circ$. Nothing stops a ray from turning further. A blade that makes one full turn and then another $45^\circ$ has turned through $360^\circ + 45^\circ = 405^\circ$. Its terminal side lies exactly where a turn of $45^\circ$ would end.

Angles with the same terminal side are called coterminal. They differ by whole turns:

So $45^\circ$, $405^\circ$, $765^\circ$ and $-315^\circ$ all end in the same place.

To find the angle from $0^\circ$ to $360^\circ$ with the same end, add or take away $360^\circ$ until the angle lies in that range. For $-45^\circ$, the answer is $-45^\circ + 360^\circ = 315^\circ$.

Quadrants

The axes split the plane into four quadrants, numbered anticlockwise from the first. An angle in standard position lies in the quadrant of its terminal side.

An angle whose terminal side lies on an axis, such as $90^\circ$ or $-180^\circ$, is a quadrantal angle. It lies in no quadrant.

Degrees, minutes and seconds

A degree is $\frac{1}{360}$ of a revolution. For finer work it is split again, as an hour is:

Example 1. Write $38^\circ 15' 36''$ in degrees. First $36'' = \frac{36}{60}' = 0.6'$, so the angle is $38^\circ 15.6'$. Then $15.6' = \frac{15.6}{60}^\circ = 0.26^\circ$. The angle is $38.26^\circ$.

Example 2. Write $71.45^\circ$ in degrees and minutes. The part after the point is $0.45^\circ = 0.45 \times 60' = 27'$. So $71.45^\circ = 71^\circ 27'$.

Surveyors and sailors still use minutes and seconds. A latitude on a map, such as $22^\circ 34'$ N, is an angle measured this way.

A worked example

Rhea's ceiling fan is switched off while one blade points east. It slows down and stops after turning clockwise through $1290^\circ$. Where does the blade point now?

1. The sign. The turn is clockwise, so the angle is $-1290^\circ$.
2. The whole turns. $1290 = 3 \times 360 + 210$. The blade made $3$ full turns and then $210^\circ$ more.
3. The same end. $-1290^\circ + 4 \times 360^\circ = 150^\circ$. The blade ends where an anticlockwise turn of $150^\circ$ from east would end.
4. The quadrant. $150^\circ$ lies between $90^\circ$ and $180^\circ$: the second quadrant. The blade points $30^\circ$ north of west.

Check: $210^\circ$ clockwise from east is the same as $360^\circ - 210^\circ = 150^\circ$ anticlockwise. Both ways agree.

Practise with new numbers

Try it in Python

The program takes five angles. For each one it gives the complete turns, the angle from $0^\circ$ to $360^\circ$ with the same end, and the quadrant. Python's % does the main work: for a whole number a, a % 360 always lies from $0$ to $359$, even when a is negative.

Common mistakes

• Forgetting the sign. A clockwise turn is negative. A turn of $90^\circ$ clockwise is $-90^\circ$, which ends like $270^\circ$, not like $90^\circ$.
• Taking away one turn only. $1000^\circ - 360^\circ = 640^\circ$ is still more than a revolution. Keep going until the angle lies from $0^\circ$ to $360^\circ$: here $280^\circ$.
• Reading minutes as hundredths. $20^\circ 30'$ is $20.5^\circ$, not $20.3^\circ$, because a degree has $60$ minutes.
• Giving a quadrantal angle a quadrant. The terminal side of $-90^\circ$ lies on the negative $y$-axis, between quadrants III and IV. It is in neither.

Remember these

• An angle is a turn from the initial side to the terminal side. Anticlockwise is positive, clockwise is negative.
• One revolution is $360^\circ$. Angles that differ by a multiple of $360^\circ$ are coterminal.
• An angle lies in the quadrant of its terminal side. An angle on an axis is quadrantal.
• $1^\circ = 60'$ and $1' = 60''$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Do $-90^\circ$ and $270^\circ$ describe the same turn? Do they end in the same place?

2. Write two positive angles and one negative angle that end where $50^\circ$ ends.

3. In which quadrant does an angle of $-500^\circ$ end?

4. Write $25.75^\circ$ in degrees and minutes.

5. How many seconds of arc make one degree?