TRIGONOMETRIC IDENTITIES - Machine Intelligence Lab · PDF fileTom Penick [email protected] ...

2
Tom Penick [email protected] www.teicontrols.com/notes 2/20/2000 TRIGONOMETRIC IDENTITIES The six trigonometric functions : sin θ= = opp hyp y r csc sin θ θ = = = hyp opp r y 1 cos θ= = adj hyp x r sec cos θ θ = = = hyp adj r x 1 tan sin cos θ θ θ = = = opp adj y x cot tan θ θ = = = adj opp x y 1 Sum or difference of two angles : sin ( ) sin cos cos sin a b a b a b ± = ± cos( ) cos cos sin sin a b a b a b ± = m tan( ) tan tan tan tan a b a b a b ± = ± 1 m Double angle formulas : tan tan tan 2 2 1 2 θ θ θ = - sin sin cos 2 2 θ θ θ = cos cos 2 2 1 2 θ θ = - cos sin 2 1 2 2 θ θ = - cos cos sin 2 2 2 θ θ θ = - Pythagorean Identities : sin cos 2 2 1 θ θ= tan sec 2 2 1 θ θ = cot csc 2 2 1 θ θ = Half angle formulas : sin ( cos ) 2 1 2 1 2 θ θ = - cos ( cos ) 2 1 2 1 2 θ θ = sin cos θ θ 2 1 2 - cos cos θ θ 2 1 2 tan cos cos sin cos cos sin θ θ θ θ θ θ θ 2 1 1 1 1 - = = - Sum and product formulas : sin cos [sin( ) sin ( )] a b a b a b = - 1 2 cos sin [sin ( ) sin ( )] a b a b a b = - - 1 2 cos cos [cos ( ) cos ( )] a b a b a b = - 1 2 sin sin [cos ( ) cos ( )] a b a b a b = - - 1 2 ( ( sin sin sin cos a b a b a b = - 2 2 2 ( ( sin sin cos sin a b a b a b - = - 2 2 2 ( ( cos cos cos cos a b a b a b = - 2 2 2 ( ( cos cos sin sin a b a b a b - =- - 2 2 2 Law of cosines : a b c bc A 2 2 2 2 = - cos where A is the angle of a scalene triangle opposite side a. Radian measure : 8.1 p420 1 180 °= π radians 1 180 radian = ° π Reduction formulas : sin( ) sin - = - θ θ cos( ) cos - = θ θ sin( ) sin( ) θ θ π = - - cos( ) cos( ) θ θ π = - - tan( ) tan - = - θ θ tan( ) tan( ) θ θ π = - ) cos( sin 2 π ± = x x m ) sin( cos 2 π ± = ± x x Complex Numbers : θ ± θ = θ ± sin cos j e j ) ( cos 2 1 θ - θ = θ j j e e ) ( sin 2 1 θ - θ - = θ j j j e e TRIGONOMETRIC VALUES FOR COMMON ANGLES Degrees Radians sin q cos q tan q cot q sec q csc q 0° 0 0 1 0 Undefined 1 Undefined 30° π/6 1/2 2 / 3 3 / 3 3 3 / 3 2 2 45° π/4 2 / 2 2 / 2 1 1 2 2 60° π/3 2 / 3 1/2 3 3 / 3 2 3 / 3 2 90° π/2 1 0 Undefined 0 Undefined 1 120° 2π/3 2 / 3 -1/2 - 3 - 3 / 3 -2 3 / 3 2 135° 3π/4 2 / 2 - 2 / 2 -1 -1 - 2 2 150° 5π/6 1/2 - 2 / 3 - 3 / 3 - 3 - 3 / 3 2 2 180° π 0 -1 0 Undefined -1 Undefined 210° 7π/6 -1/2 - 2 / 3 3 / 3 3 - 3 / 3 2 -2 225° 5π/4 - 2 / 2 - 2 / 2 1 1 - 2 - 2 240° 4π/3 - 2 / 3 -1/2 3 3 / 3 -2 - 3 / 3 2 270° 3π/2 -1 0 Undefined 0 Undefined -1 300° 5π/3 - 2 / 3 1/2 - 3 - 3 2 - 3 / 3 2 315° 7π/4 - 2 / 2 2 / 2 -1 -1 2 - 2 330° 11π/6 -1/2 2 / 3 - 3 / 3 - 3 3 / 3 2 -2 360° 2π 0 1 0 Undefined 1 Undefined

Transcript of TRIGONOMETRIC IDENTITIES - Machine Intelligence Lab · PDF fileTom Penick [email protected] ...

Page 1: TRIGONOMETRIC IDENTITIES - Machine Intelligence Lab · PDF fileTom Penick tomzap@eden.com   2/20/2000 TRIGONOMETRIC IDENTITIES The six trigonometric functions: sinθ = = opp

Tom Penick [email protected] www.teicontrols.com/notes 2/20/2000

TRIGONOMETRIC IDENTITIES

The six trigonometric functions:

sinθ = =opp

hyp

yr

cscsin

θθ

= = =hyp

opp

ry

1

cosθ = =adjhyp

xr

seccos

θθ

= = =hyp

adj

rx

1

tansin

cosθ

θθ

= = =opp

adj

yx

cottan

θθ

= = =adjopp

xy

1

Sum or difference of two angles: sin ( ) sin cos cos sina b a b a b± = ± cos( ) cos cos sin sina b a b a b± = ∓

tan( )tan tan

tan tana b

a ba b

± =±

1∓

Double angle formulas: tantantan

22

1 2θθ

θ=

sin sin cos2 2θ θ θ= cos cos2 2 12θ θ= −

cos sin2 1 2 2θ θ= − cos cos sin2 2 2θ θ θ= −

Pythagorean Identities: sin cos2 2 1θ θ+ =

tan sec2 21θ θ+ = cot csc2 21θ θ+ =

Half angle formulas:

sin ( cos )2 12

1 2θ θ= − cos ( cos )2 1

21 2θ θ= +

sincosθ θ

21

2= ±

− coscosθ θ

21

2= ±

+

tancoscos

sincos

cossin

θ θθ

θθ

θθ2

11 1

1= ±

−+

=+

=−

Sum and product formulas: sin cos [sin( ) sin ( )]a b a b a b= + + −1

2

cos sin [sin ( ) sin ( )]a b a b a b= + − −12

cos cos [cos ( ) cos ( )]a b a b a b= + + −1

2

sin sin [cos ( ) cos ( )]a b a b a b= − − +1

2 ( ) ( )sin sin sin cosa b a b a b+ = + −2 2 2

( ) ( )sin sin cos sina b a b a b− = + −2 2 2

( ) ( )cos cos cos cosa b a b a b+ = + −2 2 2

( ) ( )cos cos sin sina b a b a b− = − + −2 2 2

Law of cosines: a b c bc A2 2 2 2= + − cos where A is the angle of a scalene triangle opposite side a.

Radian measure: 8.1 p420 1180

° =π

radians

1180

radian =°

π

Reduction formulas: sin( ) sin− = −θ θ cos( ) cos− =θ θ

sin( ) sin( )θ θ π= − − cos( ) cos( )θ θ π= − −

tan( ) tan− = −θ θ tan( ) tan( )θ θ π= −

)cos(sin 2π±= xx∓ )sin(cos 2

π±=± xx

Complex Numbers: θ±θ=θ± sincos je j

)(cos 21 θ−θ +=θ jj ee )(sin 2

1 θ−θ −=θ jjj ee

TRIGONOMETRIC VALUES FOR COMMON ANGLES

Degrees Radians sin θθ cos θθ tan θθ cot θθ sec θθ csc θθ 0° 0 0 1 0 Undefined 1 Undefined

30° π/6 1/2 2/3 3/3 3 3/32 2

45° π/4 2/2 2/2 1 1 2 2

60° π/3 2/3 1/2 3 3/3 2 3/32

90° π/2 1 0 Undefined 0 Undefined 1

120° 2π/3 2/3 -1/2 - 3 - 3/3 -2 3/32

135° 3π/4 2/2 - 2/2 -1 -1 - 2 2

150° 5π/6 1/2 - 2/3 - 3/3 - 3 - 3/32 2

180° π 0 -1 0 Undefined -1 Undefined 210° 7π/6 -1/2 - 2/3 3/3 3 - 3/32 -2

225° 5π/4 - 2/2 - 2/2 1 1 - 2 - 2

240° 4π/3 - 2/3 -1/2 3 3/3 -2 - 3/32

270° 3π/2 -1 0 Undefined 0 Undefined -1

300° 5π/3 - 2/3 1/2 - 3 - 3 2 - 3/32

315° 7π/4 - 2/2 2/2 -1 -1 2 - 2

330° 11π/6 -1/2 2/3 - 3/3 - 3 3/32 -2

360° 2π 0 1 0 Undefined 1 Undefined

Page 2: TRIGONOMETRIC IDENTITIES - Machine Intelligence Lab · PDF fileTom Penick tomzap@eden.com   2/20/2000 TRIGONOMETRIC IDENTITIES The six trigonometric functions: sinθ = = opp

Tom Penick [email protected] www.teicontrols.com/notes 2/20/2000

Expansions for sine, cosine, tangent, cotangent:

3 5 7

sin 6 5! 7!y y y

y y= − + − +L

2 4 6

cos 12 4! 6!y y y

y = − + − +L

3 52

tan 3 15y y

y y= + + +L

3 51 2

cot 3 45 945y y y

yy

= − − − −L

Hyperbolic functions:

( )yy eey −−=21

sinh sinh j jsiny y=

( )yy eey −+=21

cosh cosh j jcosy y=

tanh j j tany y=

Expansions for hyperbolic functions:

L++=6

sinh 3y

yy

L++=2

1cosh 2y

y

L−+−=24

52

1sech 42 yy

y

L+−+=453

1ctnh

3yyy

y

L−+−=3607

61

csch 3yy

yy

3 52

tanh 3 15y y

y y= − + −L