Quadrants And Graphs

Quadrants And Graphs
Important trigonometric functional relationships

Example 1: sin 120° = sin (190°+30°). Here n = 1, an odd integer.
     Sin changes to Cos
      120° lies in the second quadrant  \(\Rightarrow \)Original (given) ratio sin is + ve. 
      sin 120° = sin (90°+30°) = +cos 30° = \(\sqrt 3 /2\).
Example 2: cos 240° = cos (180° + 60°) = cos (290° + 60°)
    Here  n = 2, an even integer cos remains cos. 
    240° lies in the third quadrant  original (given) ratio cos is –ve. 
     cos 240° = – cos 60° = –1/2. 
Example 3: tan (–300°) = – tan 300° = –tan (3\(\times\)90° + 30°). 
    Here n = 3, an odd integer. tan 300° lies in the fourth quadrant 
     original (given) ratio tan is –ve
     tan (–300°) = –(–cot30°) =\(\sqrt{3}\) .
The most important values of trigonometric functions.

Important Trigonometrical formulae:
Pythagorean Identities 
    1) sin² \(\theta\)+ cos²\(\theta\) = 1     2) 1+ tan² \(\theta\)= sec²\(\theta\)     3) 1+ cot²\(\theta\) = cosec² \(\theta\)
Addition formulae :
1)\(sin\left( {A + B} \right) = \sin A\cos B + \cos A\sin B\)          2) \(\cos \left( {A + B} \right) = \cos A\cos B - \sin A\sin B\)
 3) \(tan \left( {A + B} \right) = \frac{{\tan A + \tan B}}{{1 - \tan A\tan B}}\)
Subtraction formulae :
1)   \(sin\left( {A - B} \right) = \sin A\cos B - \cos A\sin B\)      
2) \(\cos \left( {A - B} \right) = \cos A\cos B + \sin A\sin B\)
3) \(\tan \left( {A - B} \right) = \frac{{\tan A - \tan B}}{{1 + \tan A\tan B}}\)
Sum and difference formulae :
1) \(\sin A + \sin B = 2\sin \left( {\frac{{A + B}}{2}} \right)\cos \left( {\frac{{A - B}}{2}} \right)\)
2) \(\sin A - \sin B = 2\cos \left( {\frac{{A + B}}{2}} \right)\sin \left( {\frac{{A - B}}{2}} \right)\)
3)  \(\cos A + \cos B = 2\cos \left( {\frac{{A + B}}{2}} \right)\cos \left( {\frac{{A - B}}{2}} \right)\)  
4)  \(\cos A - \cos B = - 2\sin \left( {\frac{{A + B}}{2}} \right)\sin \left( {\frac{{A - B}}{2}} \right)\)  \(= 2\sin \left( {\frac{{A + B}}{2}} \right)\sin \left( {\frac{{B - A}}{2}} \right)\)
Formulae related to multiple angles :
1)    sin2A = 2sinAcosA; sinA = 2sin\(A \over 2\) cos\(A \over 2\)
2)    cos2A = cos²A – sin2 A = 2cos²A –1=1–2sin² A
3)    cosA =2cos^{2\(A \over 2\)}–1=1–2sin² ; 1+cosA = 2cos² \(A \over 2\)
4)    1–cosA = 2sin² \(A \over 2\)
5)    \(\tan 2A = \frac{{2\tan A}}{{1 - {{\tan }^2}A}}\)