TRIGONOMETRY RATIOS, FUNCTIONS AND IDENTITIES

INTRODUCTION

The word trigonometry is derived from two greek words 'trigonon' and metron'. The word 'trigonon' means a triangle and the word 'metron' means a measure. Hence the word trigonometry means the study of properties of triangles. This involves the measurement of its angles and lengths.

An angle is considered as the figure obtained by rotating a given ray about its end point. The initial position OA is called the initial side and the final position OB is called terminal side of the angle. The end point O about which the ray rotates is called the vertex of the angle.

SENSE OF AN ANGLE

The sense of an angle is said to be positive or negative according as the initial side rotates in anticlockwise or clockwise direction to get to the terminal side.

SYSTEM OF MEASUREMENT OF ANGLES

There are three system for measuring angles

(1) Sexagesimal or English system: Therefore

1 right angle $= 90$ degree $( = {90^^\circ })$

${1^^\circ } = 60\,\,minutes\,\,\left( { = {{60}^\prime }} \right)$

${1^\prime } = 60{{\rm second}\nolimits} \left( { = {{60}^{\prime \prime }}} \right)$

(2) Centesimal or French system : Therefore,

1 right angle = 100 grades $( = \,\,{100^g})$

1 grade $= 100$ minutes $\left( { = {{100}^\prime }} \right)$

1 minute $= 100$ seconds $\left( { = {{100}^{\prime \prime }}} \right)$

(3) Circular system:

The measure of an angle subtended at the centre of a circle by an arc of length equal to the radius of the circle.

Consider a circle of radius $r$ having centre at $O$ . Let $A$ be a point on the circle. Now cut off an arc $AP$ whose length is equal to the radius $r$ of the circle. Then by the definition the measure of angle AOP is 1 radian $\left( { = {1^c}} \right)$

Relation between three systems of measurement of an angle :

Let $D$ be the number of degrees, R be the number of radians and G be the number of grades in an angle $\theta$ , then

$\frac{D}{{90}} = \frac{G}{{100}} = \frac{{2R}}{\pi }$

This is the required relation between the three systems of measurement of an angle.

Therefore, one radian $= \frac{{{{180}^^\circ }}}{\pi } \Rightarrow \pi$ radians $= {180^^\circ }$

i.e., 1 radian $= {57^^\circ }{17^\prime }{44.8^{\prime \prime }} \approx {57^^\circ }{17^\prime }{45^{\prime \prime }}$

Relation between an arc and an angle :

If s is the length of an arc of a circle of radius r, then the angle $\theta$ (in radians) subtended by this arc at the centre of the circle is given by $\theta = \frac{s}{r}$ or $s = r\theta$

i.e. $({\rm{Arc}} = {\rm{ radius }} \times$ angle in radian )

Sectorial area : Let OAB be a sector having central angle ${\theta ^C}$ and radius r. Then area of the sector OAB is given by $\frac{1}{2}{r^2}\theta$ .

ILLUSTRATION-1 The angle subtended at the centre of a circle of radius 3 metres by an arc of length 1 metre is equal to

(a) ${20^^\circ }$

(b) ${60^^\circ }$

(d) 3 radians

Solution : (c) Given that radius (r)= 3m and arc (d) = 1m We know that Angle $= \frac{{{{\rm arc}\nolimits} }}{{{\rm{ radius }}}} = \frac{1}{3}$ radian

ILLUSTRATION-2 A circular wire of radius $7\,\,{\rm{cm}}$ is cut and bend again into an arc of a circle of radius $12\,\,{\rm{cm}}$ . The angle subtended by the arc at the centre is

(a) ${50^^\circ }$

(b) ${210^^\circ }$

(d) ${60^^\circ }$

Solution : (b) Given that diameter of circular wire $= 14\,\,{\rm{cm}}$ Therefore length of circular wire $= 14\pi \,\,{\rm{cm}}$ Required angle $= \frac{{2\pi }}{{{\rm{ radius }}}} = \frac{{14\pi }}{{12}} = \frac{{7\pi }}{6} = \frac{7}{6}\pi \cdot \frac{{{{180}^^\circ }}}{\pi } = {210^^\circ }$

ILLUSTRATION-3 The radius of the circle whose arc of length $15\,\,{\rm{cm}}$ makes an angle of $3/4$ radian at the centre is $[{\rm{KCET}}2002]$

$\begin{array}{lllllllllllllll} {{\rm{ (a) }}}&{10\,\,{\rm{cm}}} \end{array}$

(b) $20\,\,{\rm{cm}}$

(d) $22\frac{1}{2}\,\,cm$

Solution (b) Angle $= \frac{{{{\rm arc}\nolimits} }}{{{{\rm radius}\nolimits} }} = \frac{{15}}{{(3/4)}}\,\,{\rm{cm}}$ , Radius $= 20\,\,{\rm{cm}}$

Domain and range of a frigonometrical function :

If $f:X \to Y$ is a function, defined on the set X, then the domain of the function f, written as Domf is the set of all independent variables x, for which the image f(x), is well defined element of Y, called the co-domain of

Range of $f:X \to Y$ is the set of all images f(x) which belongs to Y, l.e., Range

$f = \{ f(x) \in Y:x \in X\} \subseteq Y$

The domain and range of trigonometrical / functions are tabulated as follows:

ILLUSTRATION-1

The incorrect statement is $\quad [{\rm{MNR}}1993]$

(a) $\sin \theta = - \frac{1}{5}$

(b) $\cos \theta = 1$

(d) $\tan \theta = 20$

Solution

ILLUSTRATION-2

Which one of the following is possible [KCET 2009]

(a) $\tan \theta = 45$

(b) $\cos \theta = \frac{7}{3}$

(d) $\sec \theta = \frac{4}{5}$

Solution

(a) because $\cos \theta > 1,\,\,\sin \theta > 1$ and $\sec \theta < 1$ are not possible and $\tan \theta = 45$ is possible.

ILLUSTRATION-3

Which of the following relations is correct [WB JEE 1991]

(a) $\sin 1 < \sin {1^^\circ }$

(b) $\sin 1 > \sin {1^^\circ }$

(d) $\frac{\pi }{{180}}\sin 1 = \sin {1^^\circ }$

Solution

(b) The true relation is $\sin 1 > \sin {1^^\circ }$ since value of $\sin \theta$ is increasing $\left[ {0 \to \frac{\pi }{2}} \right]$

Trigonometrical ratios or functions

In the right angled triangle OMP, we have base $= OM = x$ perpendicular $= PM = y$ and hypotenues $= OP = r.$ We define the following trigonometric ratio which are also known as trigonometric function.