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When you first learned the meaning of the word *angle* it
was probably used to refer to the sharp corner where two lines
meet. Later you learned that this was only one kind of angle, and you
learned to distinguish between *acute*, *obtuse* and
*right* angles. Later still you encountered the strange concept
of a *straight angle* and you learned about *reflex
angles*. Now we're also including the concept of positive and
negative angles. Notice how the idea of an angle has been becoming
increasingly broad.

The diagram above illustrates the range of angular measurement, in degrees, that each stage of the growing concept of an angle includes.

In fact, there is no need to stop at plus or minus 360
degrees. Why should we stop there? You might argue that an angle of,
say, 365 degrees is no different from an angle of 5 degrees, and you'd
be right: there is no *effective* difference, but they to still
represent different amounts of turn. You end up facing in the same direction
but 365° represents a little over a full turn while 5° is a very
small turn. Similarly, the difference between an angle of 5 degrees
and the corresponding reflex angle of 355 degrees (or -355 degrees,
since we now care about in which direction the angle is measured): one is a
small turn anti-clockwise and the other is an almost complete revolution
clockwise to end up facing the same direction. In many situations we treat these
as same angle just represented in a number of different ways. This is
illustrated in the interactive below. The diagram on the left shows a negative
angle between -180 and -540 degrees. The centre one shows an angle
measuring between -180 and +180 degrees. The right diagram shows an
angle measuring between +180 and +540 degrees.

Now consider this: there is no difference between the position of a
point *P* on the unit circle for an angle \(\theta\) and the position for
angle \(\theta + n \times 360^\circ\), where \(n\) is a
positive or negative integer. And if point *P* is the same,
then the projection onto the *y-*axis must also be the
same. From this we can conclude:

The result of this is that \(\sin(\theta)\) is a *periodic
function*. The function repeats itself every 360 degrees, as shown
in the interactive below.

- What is the
*domain*of the sine function? (That is, for \(y=\sin(\theta)\) what are the minimum and maximum values \(\theta\) can take?) - Calculate \(\sin(360\,000\,036^\circ)-\sin(-324^\circ)\).
- For what values of \(\theta\) does \(\sin(\theta)\)
- reach its maximum value?
- reach its minimum value?
- intersect the horizontal axis?

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