Trigonometric Ratios of (180° + θ)

$ \displaystyle \ \ \ \text{In}\ \vartriangle PON,$

$ \displaystyle \ \ \ \sin \theta =y$

$ \displaystyle \ \ \ \cos \theta =x$

$ \displaystyle \ \ \ \tan \theta =\frac{y}{x}$

$ \displaystyle \ \ \ \cot \theta =\frac{x}{y}$

$ \displaystyle \ \ \ \sec \theta =\frac{1}{x}$

$ \displaystyle \ \ \ \operatorname{cosec}\theta =\frac{1}{y}$

$ \displaystyle \ \ \ \text{Since}\ \vartriangle {P}'O{N}'\cong \vartriangle PON,$

$ \displaystyle \ \ \ {y}'=y\ \text{and }{x}'=x\ \text{numerically}\text{.}$

$ \displaystyle \ \ \ \text{But }{P}'({x}',{y}')\ \text{lies in the third quadrant}\text{.}$

$ \displaystyle \therefore {y}'=-y\ \text{and }{x}'=-x.$

$ \displaystyle \therefore \sin (180{}^\circ +\theta )={y}'=-y=-\sin \theta $

$ \displaystyle \ \ \ \cos (180{}^\circ +\theta )={x}'=-x=-\cos \theta $

$ \displaystyle \ \ \ \tan (180{}^\circ +\theta )=\frac{{{y}'}}{{{x}'}}=\frac{{-y}}{{-x}}=\tan \theta $

$ \displaystyle \ \ \ \cot (180{}^\circ +\theta )=\frac{{{y}'}}{{{x}'}}=\frac{{-x}}{{-y}}=\cot \theta $

$ \displaystyle \ \ \ \sec (180{}^\circ +\theta )=\frac{1}{{{x}'}}=-\frac{1}{x}=-\sec \theta $

$ \displaystyle \ \ \ \operatorname{cosec}(180{}^\circ +\theta )=\frac{1}{{{y}'}}=-\frac{1}{y}=-\operatorname{cosec}\theta $

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