Trigonometry 02 (Measurements of Angles & Trigonometric Ratios)

• Definition of Angle

Let's assume that the line $OP$ is rotating in the same plane, around the point $O$ as shown in the figure. Let its origin is $OA$ and it is getting on to the positions $OB, OC, OD...$ respectively. Then we measure the angle between $OA$ and any position of $OP$ such as $OB$, using the amount of rotation that it made during its relocation from $OA$ to $OB$. According to this, there can be any number of complete rotations through $OA$, when $OP$ comes to $OB$. 

So not as in Geometry, in Trigonometry there can be any magnitude for the angles, without limitations. According to the above figure $O$ is the Origin, $OA$ is the Initial Line and $OP$ is the Rotating Line (Generating Line/ Radius Vector).

• Measurement of angles

Measuring the angles has happened in ancient history as well. For the measurement of angles we use the unit of complete revolution; that is the angle of a complete cycle which subtend the center of a circle. This complete angle has divided in to 360 equal parts and has called them degrees, by the ancient astronomers.

• Sexagesimal measurement 

In this measuring system, angle in a complete cycle has divided into four equal parts and has named them as right angles. Each right angle has divided into 90 equal parts and has introduced each part as a degree, and again each degree has divided into 60 equal parts and called each part as a minute. Finally each minute has divided again into 60 equal parts and named them as seconds.
The angle which describe as 54 degrees, 45 minutes and 3.45 seconds, can be denoted by $54^0$ $45'$ $3.45''$.

• Radian measurements

Practically we use the above mentioned sexagesimal system for measuring angles. But in theoretical applications radians are used widely for measuring angles. A radian has defined as the angle which subtend the center by an arc, having the length of the radius of the particular circle.

$AOB\measuredangle=1rad$

• The ratio between the circumference and the diameter, is a constant in any circle. 

To derive this, we have to consider the circle as a polygon with $n$ number of sides. Consider two circles with radii $r_1$ and $r_2$ and centers $O_1$ and $O_2$.

If we consider these two are regular polygons and assume the length of a side of the first polygon is $A_1 B_1$ and the length of a side of the second polygon is $A_2 B_2$.
So $A_1 O_1 B_1\measuredangle=A_2 O_2 B_2\measuredangle=\cfrac{360^0}{n}$.
Hence both triangles $A_1 O_1 B_1$ and $A_2 O_2 B_2$ become equi-angular.

$\therefore \cfrac{A_1B_1}{O_1A_1}=\cfrac{A_2B_2}{O_2A_2}$

Let $A_1B_1=l_1$ and $A_2B_2=l_2$, then,

$\cfrac{l_1}{r_1}=\cfrac{l_2}{r_2}$

$\therefore \cfrac{nl_1}{r_1}=\cfrac{nl_2}{r_2}$

If the number of sides $n$ is going to be increased infinitely, then $nl_1$ tends to be the circumference of the circle with radius $r_1$. Same thing happens to the other circle as well. If the circumferences of circles are $C_1$ and $C_2$ respectively then,

$\cfrac{C_1}{r_1}=\cfrac{C_2}{r_2}$

This shows us that the ratio between the circumference and the radius of any  circle is a constant. If so, the ratio between the circumference and the diameter of any circle is probably a constant.
$\therefore \cfrac{circumference}{diameter}$ is a constant.
This constant is represented by the Greek letter $\pi$. The value of this constant is approximately equal to the $\cfrac{22}{7}$. However exact constant is an Irrational number.

Circumference $C=2\pi r$ 

• Radians and Degrees

As we know the angle that  subtend the center of a circle by a complete cycle is $360^0$. According to the definitions of a radian,
Subtend angle by a $r$ arc length $=1$ $rad$
$\therefore$ The angle of a complete cycle $=\cfrac{1}{r}2\pi r$     $(\because C=2\pi r)$
$\therefore 360^0=2\pi$ $rad$
$1$ $rad=\cfrac{360^0}{2\pi}\approx 57^0 18'$

• Length of an arc that subtends the center by a $\theta$ angle.

To be continued...