TRIGONOMETRIC RATIOS

TRIGONOMETRIC RATIOS
Pythogoras Theorem :
    In a right-angled triangle, the sum of the squares of the two sides (other than hypotenuse) of a right-angled triangle is equal to the square of the hypotenuse side. Or in other words, the hypotenuse is the longest side of a right-angled triangle and it is opposite to the angle of 90<sup>0

Formula : \(hypotenus{e^2} = perpendicula{r^2} + bas{e^2}\)  
        or  \({c^2} = {a^2} + {b^2}\)</sup>
Note :
1. If \(A + B = {90^0}or\) 27⁰
    i) tanA.tanB = 1             ii) cotA.cotB = 1
    iii)sin²A+sin²B=1         iv) Cos²A+Cos²B=1
2. If  A+B=180⁰, then
    i) cosA + cosB = 0        ii) sinA - sinB = 0        iii) tanA + tanB = 0
3. If  A+B=360⁰, then
    i) sinA + sinB = 0        ii) cosA - cosB = 0        iii) tanA + tanB = 0
Ex :tan130⁰ .tan140⁰=1
    Since 130+140=270⁰
4. i) If \(a\cos \theta + b\sin \theta = c\) and \(a\cos \theta - b\sin \theta = k\) , then
    \({a^2} + {b^2} = {c^2} + {k^2}\)
     ii) If  \(a\sec \theta + b\tan \theta = c\) and \(a\tan \theta + b\sec \theta = k\), then
    \({a^2} - {b^2} = {c^2} - {k^2}\)
     iii) If \(a\cos ec\theta + b\cot \theta = c\) and  \(a\cot \theta + b\cos ec\theta = k\), then
     \({a^2} - {b^2} = {c^2} - {k^2}\)
5. i) \(\sin \theta + \sin \left( {\pi + \theta } \right) + \sin \left( {2\pi + \theta } \right) + ......... + \sin \left( {n\pi + \theta } \right) = \left\{ \begin{gathered} {\text{0 if n is odd}} \hfill \\ {\text{sin \theta if n is even}} \hfill \\ \end{gathered} \right.\)
    ii) \(\cos \theta + \cos \left( {\pi + \theta } \right) + \cos \left( {2\pi + \theta } \right) + ......... + \cos \left( {n\pi + \theta } \right) = \left\{ \begin{gathered} {\text{0 if n is odd}} \hfill \\ {\text{cos\theta if n is even}} \hfill \\ \end{gathered} \right.\)