TRIG.MA Measurement of Angles: Triangles, The Unit Circle, and Radian Measure (i) Acute Angles.
For
acute angles (in right triangles) $\Theta$ that are measured with degrees between 0 and 90
degrees, the sine and cosine are very useful and important
trigonometric functions.
Also, for acute right triangles the sine and cosine are most easily measured when the hypotenuse of the right triangle
has a unit measure, so $c= 1$.
These unit hypotenuse right triangles can all be considered in a cartesian
plane with the vertex $A$ located with cartesian coordinates $(0,0)$,
$C$ with coordinates $(\cos(\Theta),0)$ and the vertex $B$ with
coordinates $(\cos(\Theta),\sin(\Theta))$ on the unit circle with
equation $x^2 + y^2 =
1$.
(ii) General triangles and interior angles that measure between 0 and 180 degrees.
In any triangle, the sum of the interior angles measures a straight
angle ( 180 degrees). If all the interior angles are acute, the
triangle is called an acute triangle. If one of the angles measures 90
degrees, a right angle, the triangle is called a right triangle. If one
of the angles measures more than 90 degrees, that angle is obtuse, and
the triangle is called an obtuse triangle.
The general problem of triangles is to determine the measure of the
three line segments and the three interior angles of a given triangle.
Based on the theory of similar triangles, trigonometry allows one to solve the general triangle problem from
incomplete information using the sine and cosine functions for angles
measured between 0 and 180 degrees.
See for more on solving triangles and vector geometry.
(iii) Circle Similarity and $\pi$:
For
two circles $O, O'$ with circumferences $C, C'$; arcs cut by a central angle $a, a'$; diameters $d,d'$; and radii $r,r'$:
$\frac {C'}C=
\frac {d'}d=\frac {r'}r = \frac {a'}a$.
So
$C/d =C'/d' $ is a constant (well known), denoted by the Greek letter,$\pi$.
$\pi$ is an irrational number. In fact, $\pi$
is a transcendental number, meaning that there is no polynomial function,$ p$ with integer coefficients where $p(\pi) = 0$. These two facts are not simple to prove.
Result:
$C = \pi*d = 2 \pi * r$ (iv) Central Circular Angles and Radian Measure Central angles in a (unit) circle cut the circle in arcs.
The ratio of the arc lengths $t, t'$ is proportional to the ratio of the
corresponding angle measurements $\Theta , \Theta'$ (in
degrees).
$\Theta' /\Theta= t' /t$ or $\Theta/t = \Theta'/t'$. In a unit circle, the arc length determined by a central angle is
called the radian measure of the angle. One radian measures
the central angle that cuts an arc of length 1 on a circle with radius
1.
Also, $ \pi$ radians measures a central angle of
$180$ degrees, a straight angle, and $\pi /2$ radians
measures a central angle of $90$ degrees, a right angle.
It is customary to use the variable "$ t$" to denote the measure of an angle with radian measure.
$\Theta/180 = t /π$ where $\Theta$ is the measure of
an angle in degrees and $t$ is the measure of the same angle in
radians.
Examples of Degree and Radian Measure in Unit Circles
Entries in the table below show the corresponding measurements of
some common
angles with measure $\Theta$ degrees and measure $t$ radians. The
adjacent mapping diagram visualizes the scale correspondence between
degree and radian measures of an angle.
$\Theta$
t
360
$2\pi$
270
$3\pi/2$
180
$\pi$
90
$\pi/2$
0
0
60
$\pi/3$
45
$\pi/4$
30
$\pi/6$
In the following GeoGebra mapping diagram you can
control the angle using the x mark (in orange) on the $degrees$ axis
which is initially set at $45 $ degrees. This will change the central
angle and corresponding arc on the unit circle as well as indicate the
corresponding measure in radians for the angle.
