First, let us recall the definition of angle and degree measure of an angle.

**A recall of basic results**

In earlier classes, we have
learnt trigonometric ratios using a right triangle and proved trigonometric
identities for an acute angle. One wonders, how the distance between planets,
heights of Mountains, distance between far off objects like Earth and Sun,
heights of tall buildings, the speed of supersonic jets are measured or
calculated. Interestingly, such distances or heights are calculated applying
the trigonometric ratios which were derived for acute angles. Our aim is to
develop trigonometric functions defined for any real number and use them in all
branches of mathematics, in particular, in calculus. First, let us recall the
definition of angle and degree measure of an angle.

The angle AOB is a measure formed by two rays OA and OB sharing the common point O as shown in the Figure 3.1. The common point O is called the vertex of the angle. If we rotate the ray OA about its vertex O and takes the position OB, then OA and OB respectively are called the initial side and the
terminal side of the angle produced. An anticlockwise rotation generates a
positive angle (angle with positive sign), while a clockwise rotation generates
a negative angle (angle with negative sign).

One full anticlockwise (or
clockwise) rotation of OA back to itself is called one **complete rotation or revolution**.

There are three different systems
for measuring angles.

**(i) Sexagesimal system**

The Sexagesimal system is the most prevalent
system of measurement where a right angle is divided into 90 equal parts called
Degrees. Each degree is divided into 60 equal parts called Minutes, and each minute into
60 equal parts called
Seconds.

The symbols 1* ^{â—¦}*, 1 and 1 are used to denote a degree, a
minute and a second respectively.

**(ii) Centesimal system**

In the Centesimal system , the right angle is
divided into 100 equal parts, called Grades; each grade is subdivided into 100 Minutes, and each minute is
subdivided into 100 Seconds. The symbol 1* ^{g}* is used to denote a
grade.

**(iii) Circular system**

In the circular system , the radian measure
of an angle is introduced using arc lengths in a circle of radius *r*. Circular system is used in all branches of Mathematics and in
other applications in Science. The symbol 1* ^{c}* is used to denote 1 radian measure.

The degree is a unit of
measurement of angles and is represented by the symbol * ^{â—¦}*. In degrees, we split up one complete rotation into 360 equal
parts and each part is one degree, denoted by 1

We shall classify a pair of
angles in the following way for better understanding and usages.

Â·
Two angles that have the exact same measure are called **congruent
angles**.

Â·
Two angles that have their measures adding to 90* ^{â—¦}* are called

Â·
Two angles that have their measures adding to 180* ^{â—¦}* are called

Â·
Two angles between 0* ^{â—¦}* and 360

An angle is said to be
in standard position if its vertex is at
the origin and its initial side is along the positive *x*-axis. An angle is said to be in the first quadrant, if in the
standard position, its terminal side falls in the first quadrant. Similarly, we
can define for the other three quadrants. Angles in standard position having
their terminal sides along the *x*-axis or *y*-axis are called quadrantal angles. Thus, 0* ^{â—¦},* 90

One complete rotation
of a ray in the anticlockwise direc-tion results in an angle measuring of 360* ^{â—¦}*. By continuing the anticlockwise rotation, angles larger than 360

We know that six
ratios can be formed using the three lengths *a, b, c* of sides of a right
triangle *ABC*. Interestingly, these ratios lead to the definitions of six
basic trigonometric functions.

First, let us recall
the trigonometric ratios which are defined with reference to a right triangle.

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11th Mathematics : UNIT 3 : Trigonometry : A recall of basic results | Solved Example Problems, Exercise | Trigonometry | Mathematics

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