Definition of the Trigonometric Functions: Cheat Sheet

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  • Topic: Trigonometry, Trigonometric functions, Function
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  • Published : June 8, 2012
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Trig Cheat Sheet
Definition of the Trig Functions
Right triangle definition For this definition we assume that p 0 < q < or 0° < q < 90° . 2 Unit circle definition For this definition q is any angle. y

( x, y )
hypotenuse opposite
y 1 x

q
x

q
adjacent sin q = opposite hypotenuse adjacent cos q = hypotenuse opposite tan q = adjacent hypotenuse opposite hypotenuse sec q = adjacent adjacent cot q = opposite csc q = sin q = y =y 1 x cos q = = x 1 y tan q = x 1 y 1 sec q = x x cot q = y csc q =

Facts and Properties
Domain The domain is all the values of q that can be plugged into the function. sin q , q can be any angle cos q , q can be any angle 1ö æ tan q , q ¹ ç n + ÷ p , n = 0, ± 1, ± 2,K 2ø è csc q , q ¹ n p , n = 0, ± 1, ± 2,K 1ö æ sec q , q ¹ ç n + ÷ p , n = 0, ± 1, ± 2,K 2ø è cot q , q ¹ n p , n = 0, ± 1, ± 2,K

Period The period of a function is the number, T, such that f (q + T ) = f (q ) . So, if w is a fixed number and q is any angle we have the following periods. sin ( wq ) ® cos (wq ) ® tan (wq ) ® csc (wq ) ® sec (wq ) ® cot (wq ) ® T= T T T T T 2p w 2p = w p = w 2p = w 2p = w p = w

Range The range is all possible values to get out of the function. csc q ³ 1 and csc q £ -1 -1 £ sin q £ 1 -1 £ cos q £ 1 sec q ³ 1 and sec q £ -1 -¥ < tan q < ¥ -¥ < cot q < ¥

© 2005 Paul Dawkins

Formulas and Identities
Tangent and Cotangent Identities sin q cos q tan q = cot q = cos q sin q Reciprocal Identities 1 1 csc q = sin q = sin q csc q 1 1 sec q = cos q = cos q sec q 1 1 cot q = tan q = tan q cot q Pythagorean Identities sin 2 q + cos 2 q = 1 tan 2 q + 1 = sec 2 q 1 + cot 2 q = csc 2 q Even/Odd Formulas sin ( -q ) = - sin q csc ( -q ) = - csc q cos ( -q ) = cos q tan ( -q ) = - tan q Periodic Formulas If n is an integer. sin (q + 2p n ) = sin q tan (q + p n ) = tan q sin ( 2q ) = 2sin q cos q cos ( 2q ) = cos 2 q - sin 2 q = 2 cos 2 q - 1 = 1 - 2sin 2 q 2 tan q tan ( 2q ) = 1 - tan 2 q Degrees to Radians Formulas If x is an angle in degrees and t is an angle in radians then p t px 180t = Þ t= and x = 180 x 180 p sec ( -q ) = sec q cot ( -q ) = - cot q Half Angle Formulas 1 sin 2 q = (1 - cos ( 2q ) ) 2 1 cos 2 q = (1 + cos ( 2q ) ) 2 1 - cos ( 2q ) tan 2 q = 1 + cos ( 2q ) Sum and Difference Formulas sin (a ± b ) = sin a cos b ± cos a sin b cos (a ± b ) = cos a cos b m sin a sin b tan a ± tan b 1 m tan a tan b Product to Sum Formulas 1 sin a sin b = écos (a - b ) - cos (a + b ) ù û 2ë 1 cos a cos b = é cos (a - b ) + cos (a + b ) ù û 2ë 1 sin a cos b = ésin (a + b ) + sin (a - b ) ù û 2ë 1 cos a sin b = ésin (a + b ) - sin (a - b ) ù û 2ë Sum to Product Formulas æa + b ö æa - b ö sin a + sin b = 2sin ç ÷ cos ç ÷ è 2 ø è 2 ø æa + b ö æa - b ö sin a - sin b = 2 cos ç ÷ sin ç ÷ è 2 ø è 2 ø æa + b ö æa - b ö cos a + cos b = 2 cos ç ÷ cos ç ÷ è 2 ø è 2 ø æa + b ö æa - b ö cos a - cos b = -2sin ç ÷ sin ç ÷ è 2 ø è 2 ø Cofunction Formulas tan (a ± b ) = æp ö sin ç - q ÷ = cos q è2 ø æp ö csc ç - q ÷ = sec q è2 ø æp ö tan ç - q ÷ = cot q 2 è ø æp ö cos ç - q ÷ = sin q è2 ø æp ö sec ç - q ÷ = csc q è2 ø æp ö cot ç - q ÷ = tan q 2 è ø

csc (q + 2p n ) = csc q cot (q + p n ) = cot q

cos (q + 2p n ) = cos q sec (q + 2p n ) = sec q Double Angle Formulas

© 2005 Paul Dawkins

Unit Circle y
æ 1 3ö ç- , ÷ è 2 2 ø æ 2 2ö , ç÷ 2 2 ø è æ 3 1ö ç- , ÷ è 2 2ø

( 0,1)

p 2
2p 3
90° 120° 60°

p 3

æ1 3ö ç , ÷ ç2 2 ÷ è ø æ 2 2ö ç ç 2 , 2 ÷ ÷ è ø æ 3 1ö ç , ÷ ç 2 2÷ è ø

3p 4

p 4
45°

5p 6

135°

30° 150° 0° 360°

p 6

( -1,0 )

p 180°

0 2p

(1,0 )

x

æ 3 1ö ç - ,- ÷ 2ø è 2

7p 6

210° 225°

330° 315° 7p 300° 270° 4 5p 3p 3 2 æ
( 0,-1)

æ 2 2ö ,ç÷ 2 2 ø è

5p 4

11p 6

4p 3

240°

æ 3 1ö ç ,- ÷ è 2 2ø

æ 2 2ö ,ç ÷ 2 ø è 2

æ 1 3ö ç - ,÷ è 2 2 ø

1 3ö ç ,÷ è2 2 ø

For any ordered pair on the unit circle ( x, y ) : cos q = x and sin q = y Example æ 5p cos ç è 3 ö 1 ÷= ø 2 æ 5p sin ç è 3 3 ö ÷=2 ø

© 2005 Paul Dawkins

Inverse Trig Functions...
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