Trigonometric Functions as Projections
The period of the cosine function
- The domain of \(\cos(\theta)\) is \((-\infty,+\infty)\).
- \(\cos(360\,000\,036^\circ)-\cos(-324^\circ) = 0\).
- \(\cos(\theta)\) is at a maximum when point P is at the
right of the unit circle, i.e. at (1,0). This occurs when
\(\theta=0^\circ\). It also happens at \(\theta=-360^\circ\),
\(\theta=360^\circ\), and any other value of \(\theta\) that satisfies
\(\theta=n\times360^\circ\) for integer n.
- \(\cos(\theta)\) is at a minimum when point P is at the
left of the unit circle, i.e. at (-1,0). This occurs when
\(\theta=180^\circ\). It also happens at \(\theta=-180^\circ\),
and any other value of \(\theta\) that satisfies
\(\theta=180+n\times360^\circ\) for integer n.
- \(\cos(\theta)\) intersects the horizontal axis when point
P is at either (0,1) or (0,-1). This occurs when
\(\theta=90^\circ\) and \(\theta=270^\circ\). It also happens at \(\theta=-90^\circ\),
\(\theta=-270^\circ\), and any other value of \(\theta\) that satisfies
\(\theta=90 + n\times180^\circ\) for integer n. Note that
the graph of \(cos(\theta)\) passes through the horizontal axis twice
as frequently as it reaches a minimum or maximum.