Trigonometric Ratios of (360° – θ)

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

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

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

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

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

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

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

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

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

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

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

$\displaystyle \therefore \sin (360{}^\circ -\theta )={y}'=-y=-\sin \theta$

$\displaystyle \ \ \ \cos (360{}^\circ -\theta )={x}'=x=\cos \theta$

$\displaystyle \ \ \ \tan (360{}^\circ -\theta )=\frac{{{y}'}}{{{x}'}}=-\frac{{y}}{{x}}=-\tan \theta$

$\displaystyle \ \ \ \cot (360{}^\circ -\theta )=\frac{{{y}'}}{{{x}'}}=-\frac{{x}}{{y}}=-\cot \theta$

$\displaystyle \ \ \ \sec (360{}^\circ -\theta )=\frac{1}{{{x}'}}=\frac{1}{x}=\sec \theta$

$\displaystyle \ \ \ \operatorname{cosec}(360{}^\circ -\theta )=\frac{1}{{{y}'}}=-\frac{1}{y}=-\operatorname{cosec}\theta$

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