Introduction

There are six trigonometric ratios: - sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec). These ratios are actually the ratio of different sides of a triangle, like in right-angled triangle (triangle where any one angle is 90° degrees), the ratios are divided as below:

 \(sin\theta = {P\over H}\)  || \(cos\theta = {B\over H}\) ||   \(tan\theta = {P\over B}\).

\(cosec\theta = {H\over P}\) ||  \(sec\theta = {H\over B}\)  ||  \(cot\theta = {B\over P}\)

We should note that the ratio of sin, cos, tan etc. corresponds to the angle theta (θ).

 \(sin\theta = {1\over cosec\theta}\)    ||    \(cosec\theta= {1\over sin\theta}\) 

\(cos\theta= {1\over sec\theta}\)      ||      \(sec\theta= {1\over cos\theta}\)    

\(tan\theta= {1\over cot \theta}\)      ||     \(cot\theta= {1\over tan \theta}\) 

\(tan\theta= {sin\theta\over cos \theta}\)      ||      \(cot\theta= {cos\theta\over sin \theta}\)              

Many times, it is seen that students get confused that trigonometric ratio is limited to right-angled triangle only, also many publishers define these ratios for right-angled triangle only and bluntly speaks that the ratios are for right-angled triangle only, but these ratios are not limited to right-angled triangle only.

For any General Triangle the relation b/w angle, side and trigonometric ratio is as below:-

Law of Cosines

\(cosA= {{b^2+c^2- a^2}\over 2bc }\)

\(cosB= {{a^2+c^2- b^2}\over 2ac }\)

\(cosc= {{b^2+a^2- c^2}\over 2ab }\)

Law of Sine

 \({a\over sin\space A}={b\over sin\space B}= {c\over sin\space C}\)

Law of Tangents

 \({a\space -\space b\over a\space +\space b}={tan[{1\over2}(A\space-\space B)]\over tan [{1\over2}(A\space +\space B)]}\)

♦ From the above we can say if all the sides of a triangle are given, all the angles of the triangle can be found using cosine law, or if any two sides and an angle are given then the third side can be found.