Plugin identities
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This document provides definitions and formulas for trigonometric and hyperbolic functions including: - The six trigonometric functions defined in terms of opposite, adjacent, and hypotenuse sides of a right triangle. - Formulas for sums and differences of angles, double angle formulas, Pythagorean identities, and half angle formulas. - The law of cosines, radian measure conversions, reduction formulas, and formulas for complex numbers involving trig functions. - Tables of trigonometric values for common angles in degrees and radians. - Series expansions for trig functions, hyperbolic functions defined in terms of exponentials, and series expansions for hyperbolic functions.
![Tom Penick tomzap@eden.com www.teicontrols.com/notes 2/20/2000
TRIGONOMETRIC IDENTITIES
The six trigonometric functions:
sinθ = =
opp
hyp
y
r
csc
sin
θ
θ
= = =
hyp
opp
r
y
1
cosθ = =
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 sina b a b a b± = ±
cos( ) cos cos sin sina b a b a b± = m
tan( )
tan tan
tan tan
a b
a b
a b
± =
±
1m
Double angle formulas: tan
tan
tan
2
2
1 2
θ
θ
θ
=
−
sin sin cos2 2θ θ θ= cos cos2 2 12
θ θ= −
cos sin2 1 2 2
θ θ= − cos cos sin2 2 2
θ θ θ= −
Pythagorean Identities: sin cos2 2
1θ θ+ =
tan sec2 2
1θ θ+ = cot csc2 2
1θ θ+ =
Half angle formulas:
sin ( cos )2 1
2
1 2θ θ= − cos ( cos )2 1
2
1 2θ θ= +
sin
cosθ θ
2
1
2
= ±
−
cos
cosθ θ
2
1
2
= ±
+
tan
cos
cos
sin
cos
cos
sin
θ θ
θ
θ
θ
θ
θ2
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 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 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
±= xxm )sin(cos 2
π
±=± xx
Complex Numbers: θ±θ=θ±
sincos je j
)(cos 2
1 θ−θ
+=θ jj
ee )(sin 2
1 θ−θ
−=θ jj
j
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](https://image.slidesharecdn.com/plugin-identities-150916172424-lva1-app6892/85/Plugin-identities-1-320.jpg)
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Plugin identities
- 1. Tom Penick [email protected] www.teicontrols.com/notes 2/20/2000 TRIGONOMETRIC IDENTITIES The six trigonometric functions: sinθ = = opp hyp y r csc sin θ θ = = = hyp opp r y 1 cosθ = = 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 sina b a b a b± = ± cos( ) cos cos sin sina b a b a b± = m tan( ) tan tan tan tan a b a b a b ± = ± 1m Double angle formulas: tan tan tan 2 2 1 2 θ θ θ = − sin sin cos2 2θ θ θ= cos cos2 2 12 θ θ= − cos sin2 1 2 2 θ θ= − cos cos sin2 2 2 θ θ θ= − Pythagorean Identities: sin cos2 2 1θ θ+ = tan sec2 2 1θ θ+ = cot csc2 2 1θ θ+ = Half angle formulas: sin ( cos )2 1 2 1 2θ θ= − cos ( cos )2 1 2 1 2θ θ= + sin cosθ θ 2 1 2 = ± − cos cosθ θ 2 1 2 = ± + tan cos cos sin cos cos sin θ θ θ θ θ θ θ2 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 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 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 ±= xxm )sin(cos 2 π ±=± xx Complex Numbers: θ±θ=θ± sincos je j )(cos 2 1 θ−θ +=θ jj ee )(sin 2 1 θ−θ −=θ jj j 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
- 2. 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 1 2 4! 6! y y y y = − + − +L 3 5 2 tan 3 15 y y y y= + + +L 3 5 1 2 cot 3 45 945 y y y y y = − − − −L Hyperbolic functions: ( )yy eey − −= 2 1 sinh sinh j jsiny y= ( )yy eey − += 2 1 cosh cosh j jcosy y= tanh j jtany y= Expansions for hyperbolic functions: L++= 6 sinh 3 y yy L++= 2 1cosh 2 y y L−+−= 24 5 2 1sech 42 yy y L+−+= 453 1 ctnh 3 yy y y L−+−= 360 7 6 1 csch 3 yy y y 3 5 2 tanh 3 15 y y y y= − + −L