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