What is an Angle?
\pi
represents the ratio of a circle's
circumference to its diameter.
Therefore, a circle with a diameter of
1
has a
circumference of
\pi
.
A circle with a
radius of
1
has a diameter of
2
and a
circumference of
2\pi
.
When we refer to an angle in radians, we are actually not measuring the angle itself. We are measuring
the length of the arc of
radius 1 created using that angle.
A circle of radius
1
is called the unit circle. (In math, unit almost always means
1
.)
Sin/Cos/Tan
An arc of the unit circle can be used to create a triangle:
\c3{\text{Adjacent}} \quad\quad \c2{\text{Opposite}} \quad\quad \c1{\text{Hypotenuse}}
This triangle represents the location and length of the end of the arc.
\c3{\text{x-Coordinate}} \quad\quad \c2{\text{y-Coordinate}} \quad\quad \c1{\text{Radius}}
\sin
,
\cos
, and
\tan
can be defined using either of the above representations:
1. Using ratios of the triangle:
\begin{align*}
\sin(\c4{\theta}) &= \frac{\c2{\text{Opposite}}}{\c1{\text{Hypotenuse}}} = \c2{\text{Opposite}} \\\\
\cos(\c4{\theta}) &= \frac{\c3{\text{Adjacent}}}{\c1{\text{Hypotenuse}}} = \c3{\text{Adjacent}} \\\\
\tan(\c4{\theta}) &= \frac{\c2{\text{Opposite}}}{\c3{\text{Adjacent}}}
\end{align*}
2. Using ratios of the coordinates of the arc and the radius:
\begin{align*}
\sin(\c4{\theta}) &= \frac{\c2{\text{y-Coordinate}}}{\c1{\text{Radius}}} = \c2{\text{y-Coordinate}} \\\\
\cos(\c4{\theta}) &= \frac{\c3{\text{x-Coordinate}}}{\c1{\text{Radius}}} = \c3{\text{x-Coordinate}} \\\\
\tan(\c4{\theta}) &= \frac{\c2{\text{y-Coordinate}}}{\c3{\text{x-Coordinate}}}
\end{align*}
Notice how the ratios for
\sin
and
\cos
simplify conveniently due to us using the unit circle.
Here are the definitions of
\sin
,
\cos
, and
\tan
in terms of the unit circle:
\sin(\c4{\theta})
is the y-coordinate of the arc of the unit circle defined by \c4{\theta}
.
\cos(\c4{\theta})
is the x-coordinate of the arc of the unit circle defined by \c4{\theta}
.
\tan(\c4{\theta})
is the ratio of the y-coordinate to the x-coordinate of the arc of the unit circle defined by \c4{\theta}
.
(This definition is not simplified by the unit circle because tangent does not derive from the radius/hypotenuse.)