2.3 Trigonometric functions in different quadrants

  Basic Mathematics and Measurements

    1.0 Introduction
    2.0 Trigonometry
    3.0 Basic logarithmic functions
    4.0 Differentiation
    5.0 Integration
    6.0 Graphs
    7.0 Significant Figures
    8.0 Rounding off
    9.0 Errors
    10.0 Combination of errors
    11.0 Length Measuring Instruments
    12.0 Questions

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2.3 Trigonometric functions in different quadrants

Let, $$0^\circ < \theta < 90^\circ \quad ...\left( {{\text{Quadrant I}}} \right)$$So, $$90^\circ < 90^\circ + \theta < 180^\circ \quad ...\left( {{\text{Quadrant II}}} \right)$$$$180^\circ < 180^\circ + \theta < 270^\circ \quad ...\left( {{\text{Quadrant III}}} \right)$$$$270^\circ < 270^\circ + \theta < 360^\circ \quad ...\left( {{\text{Quadrant IV}}} \right)$$

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(A). For quadrant I

$\left( {90^\circ - \theta } \right)$	$\left( {360^\circ + \theta } \right)$
$\sin \left( {90^\circ - \theta } \right) = \cos \theta $	$\sin \left( {360^\circ + \theta } \right) = \sin \theta $
$\cos \left( {90^\circ - \theta } \right) = \sin \theta $	$\cos \left( {360^\circ + \theta } \right) = \cos \theta $
$\tan \left( {90^\circ - \theta } \right) = \cot \theta $	$\tan \left( {360^\circ + \theta } \right) = \tan \theta $

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(B). For quadrant II

$\left( {90^\circ + \theta } \right)$	$\left( {180^\circ - \theta } \right)$
$\sin \left( {90^\circ + \theta } \right) = \cos \theta $	$\sin \left( {180^\circ - \theta } \right) = \sin \theta $
$\cos \left( {90^\circ + \theta } \right) =- \sin \theta $	$\cos \left( {180^\circ - \theta } \right) = -\cos \theta $
$\tan \left( {90^\circ + \theta } \right) = -\cot \theta $	$\tan \left( {180^\circ - \theta } \right) = -\cot \theta $

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(C). For quadrant III

$\left( {180^\circ + \theta } \right)$	$\left( {270^\circ - \theta } \right)$
$\sin \left( {180^\circ + \theta } \right) = -\sin \theta $	$\sin \left( {270^\circ - \theta } \right) = -\cos \theta $
$\cos \left( {180^\circ + \theta } \right) =- \cos \theta $	$\cos \left( {270^\circ - \theta } \right) = -\sin \theta $
$\tan \left( {180^\circ + \theta } \right) = \tan \theta $	$\tan \left( {270^\circ - \theta } \right) = \cot \theta $

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(D). For quadrant IV

$\left( {270^\circ + \theta } \right)$	$\left( {360^\circ - \theta } \right)$
$\sin \left( {270^\circ + \theta } \right) = -\cos \theta $	$\sin \left( {360^\circ - \theta } \right) = -\sin \theta $
$\cos \left( {270^\circ + \theta } \right) =\sin \theta $	$\cos \left( {360^\circ - \theta } \right) = \cos \theta $
$\tan \left( {270^\circ + \theta } \right) = -\cot \theta $	$\tan \left( {360^\circ - \theta } \right) = -\tan \theta $