For every angle, \(\theta \), measured counterclockwise from the positive \(x\)-axis, we have a point on the unit circle. The coordinates of these points are functions of \(\theta \). We call them cosine and sine.

As the angle \(\theta \) changes, sine and cosine oscillate between \(-1\) and \(1\).

Both are periodic functions with period \(2\pi \). Therefore, we can examine just one wave to understand the whole function.

Generally, people pick \([0, 2\pi )\) as the interval to investigate.

We can see that sine and cosine have similar characteristics, just shifted.

Characteristics

Picture the radius spinning counterclockwise around the unit circle. The corresponding point on the unit circle will travel counterclockwise around the circle. As the point moves around the circle, its coordinates oscillates between \(-1\) and \(1\).

The sine function collects the vertical coordinates.

As the point revolves counterclockwise around the unit circle, the values of sine

• begin at \(0\) at an angle of \(0\)
• increase up to a maximum value of \(1\) at an angle of \(\frac {\pi }{2}\) (a quarter of the way around the circle)
• decrease to \(0\) at an angle of \(\pi \) (halfway way around the circle)
• continue decreasing to a minimum value of \(-1\) at an angle of \(\frac {3 \pi }{2}\) (three-quarters of the way around the circle)
• increase back to \(0\) at an angle of \(2 \pi \) (a full circle around)

The cosine function collects the horizontal coordinates.

As the point revolves counterclockwise around the unit circle, the values of cosine

• begin at a maximum value of \(1\) at an angle of \(0\)
• decrease to a value of \(0\) at an angle of \(\frac {\pi }{2}\) (a quater of the way around the circle)
• continue decreasing to a minimum value of \(-1\) at an angle of \(\pi \) (halfway around the circle)
• increasing to a value of \(0\) at an angle of \(\frac {3 \pi }{2}\) (three-quaters of the way around the circle)
• continue increasing back to \(1\) at an angle of \(2 \pi \) (a full circle around)