TRIGONOMETRIC RATIOS

TRIGONOMETRIC RATIOS
Introduction:
    The word ‘ trigonometry’ is derived from the Greek roots .. ‘tri’ means ‘three’ ; ‘gonia’ means ‘an angle’ ; and metron’ means ‘ meaning ‘ measure’. Thus ‘trigonometry’ means three angle measure. It is analytical study of a three angled geometric figure -- namely the triangle.
    Hipparchus ( 140BC ) , a Greek mathematician established the relationships between the sides and angles of a triangle. Greek trigonomentry was further developed by Hindu Mathematicians. They replaced the chords used by the Greeks by half chords of circles with given radii i.e ., the equivalent of our sine functions. The earliest such tables are in siddantas ( systems of Astronomy) of the 4th and 5th centuries A.D. Like numbers, modern trigonometry migreted from the Hindus to Europe via Arabs.
    The study of trigometry is of great importance in several fields - for example : in Surverying, Astronomy, Navigation and Engineering. In recent times, trigonometry is widely applied in many branches of Science and Engineering such as Seismology, design of electrical circuits, estimating the heights of tides in ocean etc.,
Definition of Sin\(\theta\) , cos \(\theta\) , tan\(\theta\) and their reciprocals ( for \(0 < \theta < \frac{\pi }{2}\) )
    We shall now learn the relations existing between the sides and angles of a right angled triangle.
    Consider the coordinate axes OX and OY in a plane. With O as the centre, construct a circle of raduis r. Let the terminal side of an acute angle \(\theta\) intersect the circle at P ( x,y ) as shown in figure . Draw a perpendicular PN from P to OX.
Then ON is the projection of OP on OX

From figure note that ON = x ,NP = y and OP = r
 \(\angle PON = \theta ,\angle PNO = {90^0}\)   
Hence \(\Delta PNO\) is a right angled triangle.

The following six ratios of the sides relate the angle \(\theta \) as
\( \begin{gathered} \sin \theta = \frac{{NP}}{{OP}} = \frac{{{\text{side opposite to \theta }}}}{{hypotenuse}} = \frac{y}{r} \hfill \\ \cos \theta = \frac{{ON}}{{OP}} = \frac{{{\text{side adjacent to \theta }}}}{{hypotenuse}} = \frac{x}{r} \hfill \\ \tan \theta = \frac{{NP}}{{ON}} = \frac{{{\text{side opposite to \theta }}}}{{{\text{side adjacent to \theta }}}} = \frac{y}{x} \hfill \\ \cos ec\theta = \frac{{OP}}{{NP}} = \frac{{hypotenuse}}{{{\text{side opposite to \theta }}}} = \frac{r}{y} \hfill \\ \sec \theta = \frac{{OP}}{{ON}} = \frac{{hypotenuse}}{{{\text{side adjacent to \theta }}}} = \frac{r}{x} \hfill \\ \cot \theta = \frac{{ON}}{{NP}} = \frac{{{\text{side adjacent to \theta }}}}{{{\text{side opposite to \theta }}}} = \frac{x}{y} \hfill \\ \end{gathered} \)
Note : 
1. \(\sin \theta \times \cos ec\theta = \frac{y}{r} \times \frac{r}{y} = 1\)     
     \(\cos ec\theta \) is the reciprocal of and vice-versa.     Similarly the two ratios \(\sec\theta \)  and \(\cot\theta \) are the reciprocals of respectively.
    Hence \(\cos ec\theta = \frac{1}{{\sin \theta }};\sec \theta = \frac{1}{{\cos \theta }}and\) \(\cot \theta = \frac{1}{{\tan \theta }}\)
2.      Trigonometric ratios are independent on the sides and dependent only on the magnitude of the angle.
3.     Trigonometric ratios are meaningful only when they associated with an angle like \(\theta\)