Trigonometric Functions as Projections
The period of the sine function
- The domain of sin(t) is (-infinity,+infinity;). (That is, it can take
values from -infinity to infinity. The round brackets indicate that the range
excludes both end values. In this case it has to, since infinity is
not really a number.) What this means is that we can calculate the
sine of any real number. Note, however, that your calculator may well
have difficulty with taking the sine of very large angles. If you
encounter problems, you may need to normalise the angle
before plugging it into your calculator. You can normalise an angle by
adding or subtracting whole number multiples of 360° until the
angle lies in the range (-180°, 180°], and then calculate the
sine. This is only safe to do because the sine function repeats every
360° so we can add or subtract 360° as many times as we like
without affecting the outcome.
- sin(360 000 036°)-sin(-324°) = 0.
First note that 360 000 036 = 360 000 000 + 36, and 360 000 000 is an integer multiple of
360, so sin(360 000 036°) = sin(36°).
Next, note
that -324 = 36 - 360, so sin(-324°) = sin(36°).
Make both
substitutions, and we get
sin(360 000 036°)-sin(-324°) =
sin(36°)-sin(36°) = 0
- sin(t) is at a maximum when point P is at the
top of the unit circle, i.e. at (0,1). This occurs when
t=90°. It also happens at t=-270°,
t=450°, and any other value of t that satisfies
t=90+n×360° for integer n.
- sin(t) is at a minimum when point P is at the
bottom of the unit circle, i.e. at (0,-1). This occurs when
t=270°. It also happens at t=-90°,
t=-450°, and any other value of t that satisfies
t=270+n×360° for integer n.
- sin(t) intersects the t-axis when point
P is at either (1,0) or (-1,0). This occurs when
t=0° and t=180°. It also happens at t=360°,
t=-540°, and any other value of t that satisfies
t=n×180° for integer n. Note that
the graph of sin(t) passes through the t-axis twice
as frequently as it reaches a minimum or maximum.