TRIGONOMETRIC IDENTITIES
The six trigonometric functions:
opp y
hyp r
1
=
= =
sin θ =
csc θ =
hyp r
opp y sin θ
adj x
=
hyp r
opp y sin θ
tan θ =
= =
adj x cos θ
cos θ =
sec θ =
cot θ =
sin 2θ = 2 sin θ cos θ
cos 2θ = 1 − 2 sin 2 θ
Pythagorean Identities:
tan θ + 1 = sec θ
2
2
cos a sin b
= 12 [sin (a + b ) − sin (a − b )]
hyp r
1
= =
adj x cos θ
cos a cos b =
1
[cos (a
2
+ b ) + cos (a − b )]
adj x
1
= =
opp y tan θ
sin a sin b =
1
[cos (a
2
− b ) − cos (a + b )]
sin a + sin b = 2 sin( a +2 b ) cos( a −2 b )
Sum or difference of two angles:
sin (a ± b ) = sin a cos b ± cos a sin b
cos(a ± b ) = cos a cos b m sin a sin b
tan a ± tan b
tan( a ± b) =
1 m tan a tan b
Double angle formulas:
Sum and product formulas:
sin a cos b = 1 [sin (a + b ) + sin (a − b )]
2
sin a − sin b = 2 cos( a +2 b ) sin( a −2 b )
cos a + cos b = 2 cos( a +2 b ) cos( a −2 b )
cos a − cos b = −2 sin( a +2 b ) sin( a −2 b )
2 tan θ
tan 2θ =
1 − tan 2 θ
cos 2θ = 2 cos 2 θ − 1
cos 2θ = cos 2 θ − sin 2 θ
sin 2 θ + cos 2 θ = 1
cot 2 θ + 1 = csc 2 θ
Half angle formulas:
1
1
cos2 θ = (1 + cos 2θ)
sin 2 θ = (1 − cos 2θ )
2
2
θ
1 − cos θ
θ
1 + cos θ
sin = ±
cos = ±
2
2
2
2
θ
1 − cos θ
sin θ
1 − cos θ
tan = ±
=
=
2
1 + cos θ 1 + cos θ
sin θ
2
2
2
Law of cosines:
a = b + c − 2bc cos A
where A is the angle of a scalene triangle opposite
side a.
π
Radian measure: 8.1 p420 1° =
radians
180
180°
1 radian =
π
Reduction formulas:
sin( −θ) = − sin θ
cos( −θ) = cos θ
sin(θ) = − sin(θ − π)
cos(θ) = − cos(θ − π)
tan( −θ) = − tan θ
tan(θ) = tan(θ − π )
m sin x = cos( x ± )
π
2
Complex Numbers:
cos θ = 12 ( e jθ + e − jθ )
± cos x = sin( x ± π2 )
e ± jθ = cos θ ± j sin θ
sin θ = j12 ( e jθ − e − jθ )
TRIGONOMETRIC VALUES FOR COMMON ANGLES
Degrees
Radians
sin θ
cos θ
0°
30°
0
π/6
0
1/2
1
45°
π/4
60°
π/3
2 /2
3/2
90°
120°
π/2
2π/3
135°
3π/4
3/2
2 /2
150°
5π/6
1/2
180°
210°
225°
π
7π/6
5π/4
0
-1/2
- 3/2
-1
- 3/2
240°
4π/3
- 2 /2
- 3/2
- 2 /2
-1/2
-1
0
1/2
1
270°
300°
3π/2
5π/3
315°
7π/4
- 3/2
- 2 /2
11π/6
2π
-1/2
0
330°
360°
3/2
2 /2
1/2
tan θ
0
3/3
1
3
0
-1/2
Undefined
- 3
- 2 /2
-1
- 3/3
0
3/3
1
2 /2
3/2
1
cot θ
sec θ
csc θ
Undefined
3
1
2 3/3
Undefined
2
1
2
2
3/3
0
- 3/3
-1
- 3
Undefined
3
1
Undefined
-2
2
2 3/3
1
2 3/3
- 2
-2 3 / 3
-1
-2 3 / 3
- 2
Undefined
-2
- 2
2
2
-2
-2 3 / 3
Undefined
- 3
0
- 3
Undefined
2
-1
- 3/3
0
-1
- 3
Undefined
2
-1
-2 3 / 3
- 2
2 3/3
1
-2
Undefined
3
Tom Penick
3/3
tomzap@eden.com
www.teicontrols.com/notes
2/20/2000
Expansions for sine, cosine, tangent, cotangent:
y3 y5 y 7
sin y = y − + − +L
6 5! 7!
y2 y4 y6
cos y = 1 − + − +L
2 4! 6!
y3 2 y5
tan y = y + +
+L
3 15
1 y y3 2 y5
cot y = − − −
−L
y 3 45 945
Hyperbolic functions:
(
)
sinh jy = jsin y
(
)
cosh jy = jcos y
1 y
e − e− y
2
1
cosh y = e y + e − y
2
sinh y =
tanh jy = j tan y
Expansions for hyperbolic functions:
y3
sinh y = y +
+L
6
y2
cosh y = 1 +
+L
2
y2 5y 4
sech y = 1 −
+
−L
2
24
1 y y3
ctnh y = + −
+L
y 3 45
1 y 7 y3
csch y = − +
−L
y 6 360
y3 2 y5
tanh y = y − +
−L
3 15
Tom Penick
tomzap@eden.com
www.teicontrols.com/notes
2/20/2000