Angles and Trigonometry

Mathematicians normally measure angles in two ways, or systems: degrees and radians. Degrees are easy and familiar. There are 360 degrees in a single rotation. A right angle is therefore 90^o. Radians are less familiar, but you must get used to them because most of mathematical theory involving angles is expressed in terms of radians.

The definition of a radian is that it is the angle subtended at the centre of a circle by a piece of the circumference of the circle of length equal to the radius of the circle.

This means that, since the circumference of a circle of radius r is 2$\pi$r, a single rotation is 2$\pi$ radians.

Outside mathematics the use of degrees is universal, because the numbers are nicer. Degrees are also the older system, being in use by Greek mathematicians in the second century BC, with throwbacks from there to the Babylonian astronomers and mathematicians with their base 60 number system. Radians, along with the trigonometric functions such as sine, were introduced by Indian mathematicians in the sixth century AD. The numbers involved are more awkward, but the system gives a neater relationship between angle and arc length on the circle, and this makes for tidier formulas elsewhere. This is especially true of calculus formulas; calculus using degrees is a mess, and so in this context radians are preferred by everyone.

When you are doing calculus you always use radians. A lot of the formulas for standard derivatives and integrals are false if you don't.

Warning: Most calculators wake up in degree mode. You must switch to radians before doing any calculation for this course which involves a trig function. If you don't, you will get wrong answers, because the formulas you are using are built on radian measurement.

We have the following obvious conversions:

360^o = 2$$\pi$$ radians    180^o = $$\pi$$ radians    90^o = $${\textstyle\tfrac{1}{2}}$$$$\pi$$ radians

45^o = $${\frac{{1}}{{4}}}$$$$\pi$$ radians    60^o = $${\frac{{1}}{{3}}}$$$$\pi$$ radians    30^o = $${\frac{{1}}{{6}}}$$$$\pi$$ radians

Angles are traditionally measured anticlockwise. An angle measured in the clockwise direction is taken to be negative (thinking of it as a `turn' rather than a `separation').