This document provides information on trigonometric identities and formulas including:
- The six trigonometric functions and their definitions 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 involving complex numbers.
- A table of trigonometric values for common angles in degrees and radians.
- Expansions for trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions.
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The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
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1. 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
2. Tom Penick tomzap@eden.com 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
![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/identities-180829184328/85/Identities-1-320.jpg)
