Trig Identities

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

[email protected]

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

[email protected]

www.teicontrols.com/notes

2/20/2000