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.

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