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Olga-lednichenko-calculus-algebra-trigonometry-pdf-trig cheat-sheet_free-download
1.
Trig Cheat Sheet
Formulas and Identities Tangent and Cotangent Identities Half Angle Formulas sin q cos q 1 Definition of the Trig Functions tan q = cot q = sin 2 q = (1 - cos ( 2q ) ) Right triangle definition cos q sin q 2 For this definition we assume that Reciprocal Identities 1 Unit circle definition 1 1 cos 2 q = (1 + cos ( 2q ) ) p For this definition q is any angle. csc q = sin q = 2 0 < q < or 0° < q < 90° . 2 y sin q csc q 1 - cos ( 2q ) 1 1 tan q = 2 sec q = cos q = 1 + cos ( 2q ) ( x, y ) cos q sec q Sum and Difference Formulas 1 1 1 hypotenuse y q cot q = tan q = sin (a ± b ) = sin a cos b ± cos a sin b opposite tan q cot q x x Pythagorean Identities cos (a ± b ) = cos a cos b m sin a sin b q sin 2 q + cos 2 q = 1 tan a ± tan b tan (a ± b ) = adjacent tan q + 1 = sec q 2 2 1 m tan a tan b Product to Sum Formulas opposite hypotenuse y 1 1 + cot 2 q = csc 2 q sin q = csc q = sin q = =y csc q = 1 sin a sin b = é cos (a - b ) - cos (a + b ) ù hypotenuse opposite 1 y Even/Odd Formulas 2ë û adjacent hypotenuse x 1 sin ( -q ) = - sin q csc ( -q ) = - csc q 1 cos q = secq = cos q = = x sec q = cos a cos b = é cos (a - b ) + cos (a + b ) ù hypotenuse adjacent 1 x cos ( -q ) = cos q sec ( -q ) = sec q 2ë û opposite adjacent y x 1 tan q = cot q = tan q = cot q = tan ( -q ) = - tan q cot ( -q ) = - cot q sin a cos b = ésin (a + b ) + sin (a - b ) ù adjacent opposite x y 2ë û Periodic Formulas 1 Facts and Properties cos a sin b = ésin (a + b ) - sin (a - b ) ù 2ë û If n is an integer. Domain sin (q + 2p n ) = sin q csc (q + 2p n ) = csc q Sum to Product Formulas The domain is all the values of q that Period cos (q + 2p n ) = cos q sec (q + 2p n ) = sec q æa + b ö æa - b ö can be plugged into the function. The period of a function is the number, sin a + sin b = 2 sin ç ÷ cos ç ÷ T, such that f (q + T ) = f (q ) . So, if w tan (q + p n ) = tan q cot (q + p n ) = cot q è 2 ø è 2 ø sin q , q can be any angle is a fixed number and q is any angle we Double Angle Formulas æa + b ö æa - b ö sin a - sin b = 2 cos ç ÷ sin ç ÷ cos q , q can be any angle have the following periods. è 2 ø è 2 ø æ 1ö sin ( 2q ) = 2sin q cos q tan q , q ¹ ç n + ÷ p , n = 0, ± 1, ± 2,K æa + b ö æa - b ö 2p cos ( 2q ) = cos 2 q - sin 2 q cos a + cos b = 2 cos ç ÷ cos ç ÷ è 2ø sin ( wq ) ® T = è 2 ø è 2 ø csc q , q ¹ n p , n = 0, ± 1, ± 2,K w = 2 cos 2 q - 1 æa + b ö æa - b ö æ 1ö 2p cos a - cos b = -2 sin ç ÷ sin ç ÷ sec q , q ¹ ç n + ÷ p , n = 0, ± 1, ± 2,K cos (wq ) ® T = = 1 - 2sin 2 q è 2 ø è 2 ø è 2ø w 2 tan q Cofunction Formulas cot q , q ¹ n p , n = 0, ± 1, ± 2,K tan (wq ) ® T = p tan ( 2q ) = 1 - tan 2 q æp ö æp ö w sin ç - q ÷ = cos q cos ç - q ÷ = sin q Range 2p Degrees to Radians Formulas è2 ø è2 ø csc (wq ) ® T = The range is all possible values to get w If x is an angle in degrees and t is an æp ö æp ö csc ç - q ÷ = sec q sec ç - q ÷ = csc q out of the function. 2p angle in radians then è2 ø è2 ø -1 £ sin q £ 1 csc q ³ 1 and csc q £ -1 sec (wq ) ® T = p t px 180t w = Þ t= and x = æp ö æp ö tan ç - q ÷ = cot q cot ç - q ÷ = tan q -1 £ cos q £ 1 sec q ³ 1 and sec q £ -1 p 180 x 180 p è 2 ø è 2 ø -¥ < tan q < ¥ -¥ < cot q < ¥ cot (wq ) ® T = w © 2005 Paul Dawkins © 2005 Paul Dawkins
2.
Unit Circle
Inverse Trig Functions Definition Inverse Properties y ( 0,1) y = sin - 1 x is equivalent to x = sin y cos ( cos -1 ( x ) ) = x cos -1 ( cos (q ) ) = q p y = cos - 1 x is equivalent to x = cos y sin ( sin -1 ( x ) ) = x sin -1 ( sin (q ) ) = q æ1 3ö æ 1 3ö ç , ÷ ç2 2 ÷ ç- , ÷ 2 è ø y = tan x is equivalent to x = tan y -1 è 2 2 ø p æ 2 2ö tan ( tan - 1 ( x ) ) = x tan -1 ( tan (q ) ) = q 2p 90° ç 2 , 2 ÷ ç ÷ æ 2 2ö 3 è ø ç- , ÷ 3 Domain and Range è 2 2 ø 120° 60° p Alternate Notation 3p æ 3 1ö Function Domain Range 4 ç 2 ,2 ÷ ç sin -1 x = arcsin x 4 è ÷ ø p p æ 3 1ö ç- , ÷ 135° 45° p y = sin -1 x -1 £ x £ 1 - £ y£ è 2 2ø 5p 2 2 cos - 1 x = arccos x 6 y = cos x -1 -1 £ x £ 1 0£ y £p 6 30° tan - 1 x = arctan x 150° p p y = tan -1 x -¥ < x < ¥ - < y< 2 2 ( -1,0 ) p 180° 0° 0 (1,0 ) x Law of Sines, Cosines and Tangents 360° 2p c b a 210° 7p 330° 11p 6 225° æ 3 1ö 6 æ 3 1ö a g ç - ,- ÷ 5p 315° ç ,- ÷ è 2 2ø è 2 2ø 4 240° 300° 7p æ 2 2ö 4p 270° 4 b ç- ,- ÷ 5p æ 2 2ö è 2 2 ø 3 3p ç ,- ÷ 3 è 2 2 ø 2 Law of Sines Law of Tangents æ 1 3ö ç - ,- ÷ æ1 ç ,- 3ö ÷ sin a sin b sin g = = a - b tan 1 (a - b ) è 2 2 ø è2 2 ø = 2 ( 0,-1) a b c a + b tan 1 (a + b ) 2 Law of Cosines b - c tan 1 ( b - g ) = 2 a 2 = b 2 + c 2 - 2bc cos a b + c tan 1 ( b + g ) 2 For any ordered pair on the unit circle ( x, y ) : cos q = x and sin q = y b 2 = a 2 + c 2 - 2ac cos b a - c tan 1 (a - g ) = 2 c = a + b - 2ab cos g a + c tan 1 (a + g ) 2 2 2 2 Example Mollweide’s Formula æ 5p ö 1 æ 5p ö 3 cos ç ÷= sin ç ÷=- a + b cos 1 (a - b ) è 3 ø 2 è 3 ø 2 = 2 c sin 1 g 2 © 2005 Paul Dawkins © 2005 Paul Dawkins
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