Trigonometric functions of negative angles





Trigonometric functions of negative angles:

 

let \theta and - \theta be any two angles equal in magnitude but opposite in sign.

If we place each of them in the standard position it can be observed that the two angles are symmetrically placed on either side of x-axis.

Suppose we construct a circle of radius “r” with centre “o” , it will cut the terminal arms of angles \theta and - \theta

as shown in figure below:

Trigonometric functions of negative angles

Trigonometric functions of negative angles

 

Let thee points “P” and “P’” respectively. Clearly , the abscissa of P is the same as that of P’ , both in magnitude and

direction. But the ordinates of P and P’ are equal in magnitude but opposite in sign. If we denote the point

P by P(x,y) , then the point P’ will have the co-ordinates P’(x,-y), thus we have:

\sin (-\theta) = \frac{-y}{r} = - \frac{y}{r} = - \sin \theta

And:

\cos ( - \theta ) = \frac{x}{r} = \cos \theta

 

Similarly:

\tan (- \theta ) = - \tan \theta and \cot (- \theta ) = - \cot \theta

 

Thus:

\sin( -\theta ) = - \sin \theta , \cos ( -\theta ) = \cos \theta , \tan (- \theta ) = - \tan \theta and \cot (- \theta ) = - \cot \theta

 



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