The Elements of Plane and Spherical Trigonometry |
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Page 23
... sin sin ( 2n + 0 ) , by Article ( 24 ) : = and sin 0 = − sin ( − 0 ) = − sin ( 2nπ - 0 ) , by Article ( 31 ) : = - - that is , sin 0 = ± sin ( 2nπ0 ) . Again , sin 0 = sin ( π – 0 ) , by Article ( 32 ) , - · sin ( 2nπ + π − 0 ) = sin ...
... sin sin ( 2n + 0 ) , by Article ( 24 ) : = and sin 0 = − sin ( − 0 ) = − sin ( 2nπ - 0 ) , by Article ( 31 ) : = - - that is , sin 0 = ± sin ( 2nπ0 ) . Again , sin 0 = sin ( π – 0 ) , by Article ( 32 ) , - · sin ( 2nπ + π − 0 ) = sin ...
Page 28
John Hind. ( 2 ) Sin 135 ° = sin ( 90 ° + 45o ) = sin 90 ° cos 45 ° 1 1 + cos 90 ° sin 45o = 1. +0 . = √2 1 √2 √2 = sin 45 ° : cos 135 ° = cos ( 90 ° + 45 ° ) = cos 90 ° cos 45o – sin 90o sin 45o 1 = 0 . - 1 . √2 √2 1 = - cos 45o ...
John Hind. ( 2 ) Sin 135 ° = sin ( 90 ° + 45o ) = sin 90 ° cos 45 ° 1 1 + cos 90 ° sin 45o = 1. +0 . = √2 1 √2 √2 = sin 45 ° : cos 135 ° = cos ( 90 ° + 45 ° ) = cos 90 ° cos 45o – sin 90o sin 45o 1 = 0 . - 1 . √2 √2 1 = - cos 45o ...
Page 29
John Hind. sin ( A - B ) = sin { ( 180 ° - A ) + B } = sin ( 180 ° - A ) cos B + cos ( 180 ° – A ) sin B = sin A cos B - cos A sin B : - cos ( A + B ) = sin { ( 90 ° — A ) – B } = sin ( 90 ° – A ) cos B - - cos ( 90 ° - A ) sin B = cos A ...
John Hind. sin ( A - B ) = sin { ( 180 ° - A ) + B } = sin ( 180 ° - A ) cos B + cos ( 180 ° – A ) sin B = sin A cos B - cos A sin B : - cos ( A + B ) = sin { ( 90 ° — A ) – B } = sin ( 90 ° – A ) cos B - - cos ( 90 ° - A ) sin B = cos A ...
Page 30
... sin ( A + B ) sin ( A – B ) = ( sin A cos B + cos A sin B ) ( sin A cos B- cos A sin B ) sin A cos B - cos A sin B = = sin3A ( 1 sin B ) - ( 1 - sin A ) sin B = sin2A - sin A sin'B - sin'B + sin2A sin2B = sin3A - sin'B = ( sin A + sin B ) ...
... sin ( A + B ) sin ( A – B ) = ( sin A cos B + cos A sin B ) ( sin A cos B- cos A sin B ) sin A cos B - cos A sin B = = sin3A ( 1 sin B ) - ( 1 - sin A ) sin B = sin2A - sin A sin'B - sin'B + sin2A sin2B = sin3A - sin'B = ( sin A + sin B ) ...
Page 32
... sin A sin 1+ cos A 2 = 1 - tan2A 1 + tan2 A and these formulę being rational , are frequently very convenient in practice . 47. COR . 3. Hence , sin 34 sin ( 24 + A ) = = sin 2A cos A + cos 2A sin A = 2 sin A cos3A + ( 1 − 2 sin3A ) sin A ...
... sin A sin 1+ cos A 2 = 1 - tan2A 1 + tan2 A and these formulę being rational , are frequently very convenient in practice . 47. COR . 3. Hence , sin 34 sin ( 24 + A ) = = sin 2A cos A + cos 2A sin A = 2 sin A cos3A + ( 1 − 2 sin3A ) sin A ...
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algebraical angular appears applied arithmetical Article assuming base called Cambridge centre chapter circle circular circumscribed cloth College contained corresponding cos a cos cos² cosec determined distance drawn easily Edition equal equation evidently expressed factors figures follows formulę functions given gives greater Hence imaginary immediately inscribed latter less lines logarithms magnitude means measure meet nearly negative object observed obtained opposite perpendicular places plane polygon positive practical produced prove quadrant quantity radius regular respectively rest right angles roots sides signs similarly sin a sin sin² sin³ sine and cosine solution sphere spherical triangle square surface tables Theorem Trigonometry values vers whence
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