Trig Cheat Sheet - Summary Trigonometry PDF

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Summaries of Trigonometry's Laws...

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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

y

1

opposite

q x

x

q adjacent opposite hypotenuse adjacent cosq = hypotenuse opposite tan q = adjacent sin q =

hypotenuse opposite hypotenuse secq = adjacent adjacent cotq = opposite cscq =

y =y 1 x cos q = = x 1 y tan q = x

sin q =

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. sinq , q can be any angle cosq , 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 secq , q ¹ ç n + ÷ p , n = 0, ± 1, ± 2,K 2ø è cotq , q ¹ n p , n = 0, ± 1, ± 2,K

Range The range is all possible values to get out of the function. cscq ³ 1 and cscq £ -1 -1 £ sinq £ 1 -1 £ cosq £ 1 secq ³ 1 and secq £ -1 -¥ < tan q < ¥ -¥ < cot q < ¥

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. 2p w 2p = w p = w 2p = w 2p = w p = w

sin (w q ) ®

T=

cos ( w q ) ®

T

tan ( w q ) ®

T

csc (w q ) ®

T

sec (w q ) ®

T

cot (w q ) ®

T

Formulas and Identities Tangent and Cotangent Identities sin q cos q tan q = cot q = cosq sin q Reciprocal Identities 1 1 csc q = sin q = sin q csc q 1 1 secq = cosq = cos q sec q 1 1 cotq = tanq = tan q cot q Pythagorean Identities 2 2 sin q + cos q = 1 tan 2 q + 1 = sec 2 q 2 2 1 + cot q = csc q

Even/Odd Formulas sin( - q ) = - sin q csc (- q ) = - csc q cos( -q ) = cosq

sec( -q ) = secq

tan ( -q ) = - tan q

cot ( -q ) = - cot q

Periodic Formulas If n is an integer. sin( q + 2p n ) = sin q

csc (q + 2p n ) = csc q

cos( q + 2pn ) = cos q sec( q + 2pn ) = sec q tan (q +p n) = tanq cot (q + p n) = cot q Double Angle Formulas sin ( 2q ) = 2 sinq cosq cos( 2q ) = cos2 q - sin2 q = 2 cos 2 q - 1 = 1- 2sin2 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 px t 180t and x = = Þ t= p 180 x 180

Half Angle Formulas (alternate form) 1- cos q 1 q sin = ± sin 2 q = (1 - cos ( 2q )) 2 2 2 cos

q 1 + cos q =± 2 2

2 cos q =

1 (1 + cos ( 2q ) ) 2 1- cos ( 2q )

q 1 - cos q tan 2 q = =± 2 1 + cosq 1 + cos( 2q ) Sum and Difference Formulas sin (a ± b ) = sin a cos b ± cos a sin b tan

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 = éë cos ( a - b ) - cos ( a + b )ùû 2 1 cosa cos b = éëcos (a - b ) + cos (a + b ) ùû 2 1 sin a cos b = éësin ( a + b ) + sin ( a - b )ùû 2 1 cosa sin b = éësin (a + b ) - sin (a - b )ùû 2 Sum to Product Formulas æ a + bö æ a - bö sin a + sin b = 2 sin ç cos ç ÷ ÷ è 2 ø è 2 ø tan ( a ± b) =

a + b ö æa - b ö sin a - sin b = 2 cos æç ÷ sin ç ÷ è 2 ø è 2 ø æ a+ bö æ a- b ö cosa + cos b = 2 cos ç cos ç ÷ ÷ è 2 ø è 2 ø æa + b cosa - cos b = -2 sin ç è 2 Cofunction Formulas

ö æa - b ö ÷sin ç 2 ÷ ø è ø

æp ö sin ç - q ÷ = cos q 2 è ø æp ö csc ç -q ÷ = sec q 2 è ø

æp ö cos ç - q ÷ = sin q 2 è ø æp ö sec ç -q ÷ = cscq è2 ø

p tan æç -q ö÷ = cot q è2 ø

p cot æç -q ö÷ = tan q è2 ø

Unit Circle y

æ 3 1ö ç- , ÷ è 2 2ø

2p 3 120°

3p 4

æ1 3 ö çç , ÷÷ è2 2 ø

p 2

æ 1 3ö ç- , ÷ è 2 2 ø æ 2 2 ö , ç÷ è 2 2 ø

(0,1 )

p 3

90°

p 4

60 ° 45°

135°

5p 6

æ 2 2 ö , çç 2 2 ÷÷ è ø

30°

p 6

æ 3 1ö , çç 2 2 ÷÷ è ø

150° ( - 1,0)

p 180°

æ 3 1ö ç- , - ÷ è 2 2ø

7p 6

æ 2 2ö ,ç÷ 2 2 è ø

210°

0°

0

360°

2p

330° 225°

5p 4

240°

4p 3

æ 1 3ö ç- , ÷ è 2 2 ø

270° 3p 2

315° 7p 300° 4 5p 3

(0, -1)

æ1 3ö ç ,- ÷ 2 2 è ø

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

3 æ 5p ö sin ç ÷ = 2 è 3 ø

11p 6

(1,0)

æ 3 1ö ,- ÷ ç è 2 2ø

æ 2 2ö ,ç ÷ 2 ø è 2

x

Inverse Trig Functions Inverse Properties cos (cos -1 ( x )) = x cos - 1 ( cos (q

Definition y = sin -1 x is equivalent to x = sin y -1

y = cos x is equivalent to x = cos y y = tan -1 x is equivalent to x = tan y Domain and Range Function Domain y = sin- 1 x

-1 £ x £ 1

y = cos-1 x

-1 £ x £ 1

y = tan -1 x

-¥ < x < ¥

sin ( sin -1 (x )) = x

sin - 1 ( sin (q )) = q

tan ( tan -1 ( x )) = x

tan -1 (tan (q ) ) = q

Alternate Notation 1 sin - x = arcsin x

Range

p p £y£ 2 2 0 £ y £p p p -...