FUNCTIONS OF ANGLES Machinery's Handbook, 31st Edition
97
Reciprocal Identities: A tan = = Negative-Angle Identities: A – ( ) sin A = –sin A sin A cos ------ 1 A cot ------
1 A cos = ------
1 A sin = ------
A sec
A csc
A – ( ) cos
A – ( ) tan
A = cos
A = –tan
Cofunction Identities: 90 A – ( ) sin
90 A – ( ) tan 90 A – ( ) cot
90 A – ( ) sec 90 A – ( ) csc
A = cos A = sin
A = cot A = tan
A = csc A = sec
90 A – ( ) cos
Pythagorean Identities: A sin 2 A cos 2 + = 1
A tan 2 +
A sec 2 =
A cot 2 +
A csc 2 =
1
1
Sum- and Difference-of-Angles Formulas: A B + ( ) sin A sin B cos A cos B sin + =
A B – ( ) sin A B – ( ) cos
=
A sin
B cos
–
A cos
B sin
A B + ( ) cos
=
A cos A tan
B cos
A sin B – sin
=
A cos
B cos
A sin B + sin
B + tan A tan B – tan = ----------------- 1 B cot – 1 A + cot = ----------------- A cot B cot
A tan B – tan A tan B + tan = ----------------- 1 A cot B cot +1 A – cot = ----------------- B cot
A B + ( ) tan
A B – ( ) tan
A B + ( ) cot
A B – ( ) cot
Double-Angle Formulas: 2 A cos A cos 2 = ( )
A sin 2 – 2 =
A cos 2 – 1 1 2 =
A sin 2 –
2
2 A tan A tan 2 – ------------ 1
2 A sin
2 A sin A cos = ( )
A cot A – tan ---------------
2 A tan ( )
=
=
Half-Angle Formulas: sin( A /2)
(1 – cos A )/2 = ± cos( A /2)
(1 + cos A )/2 = ±
A – cos A sin -----------
A sin A + cos ----------- 1
1 1 A – cos A + cos ----------- 1 =
tan( A /2) = ± Product-to-Sum Formulas: A sin B cos 1 ⁄ 2 [ =
=
A B + ( ) sin
A B – ( ) + sin
]
1 ⁄
cos [
A B + ( )
A B – ( ) + cos
]
A cos
B cos
=
2
1 ⁄
cos [
A B – ( )
A B + ( ) – cos
]
A sin
B sin
=
2
A tan A cot
B + tan
A tan B + cot = --------------- Sum and Difference of Functions Formulas: B tan
A B + ( ) sin A cos B cos = ------------- A B – ( ) sin A cos B cos = ------------- B A + ( ) sin A sin B sin = ------------- B A – ( ) sin A sin B sin = -------------
A B – 2 -------
A B + 2 ------- A B – 2 -------
A tan
B + tan
A sin
B + sin 2 =
cos
sin
A B + 2 -------
A tan
B – tan
A sin
B – sin 2 =
cos
sin
A B – ( ) 2 cos ---------
A B + 2 -------
A cot
B + cot
A cos
B + cos
2
=
cos
A B – 2 -------
A B + 2 -------
A cot
B – cot
A cos
B – cos
=
2
sin
sin
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