Mathematical Formula

MATHEMATICAL FORMULA Quadratic Formula • The roots of the quadratic equation ax 2 + bx + c = 0 are

x1 , x 2 =

− b ± b 2 − 4ac 2a

Trigonometric Identities 1. sin (− x ) = − sin x 2. cos(− x ) = cos x sin x 3. tan x = cos x 1 4. cot x = tan x 1 5. csc x = sin x 1 6. sec x = cos x 7. sin x ± 90 o = ± cos x 8. cos x ± 90 o = m sin x 9. sin x ± 180 o = − sin x 10. cos x ± 180 o = − cos x 11. cos 2 x + sin 2 x = 1 12. 1 + tan 2 x = sec 2 x 13. cot 2 x + 1 = csc 2 x 14. sin( x ± y ) = sin x cos y ± cos x sin y 15. cos( x ± y ) = cos x cos y m sin x sin y tan x ± tan y 16. tan ( x ± y ) = 1 m tan x tan y 17. sin 2 x = 2 sin x cos x 18. cos 2 x = cos 2 x − sin 2 x = 1 − 2 sin 2 x = 2 cos 2 x − 1 2 tan x 19. tan 2 x = 1 − tan 2 x 1 20. sin 2 x = (1 − cos 2 x ) 2 1 21. cos 2 x = (1 + cos 2 x ) 2 1 22. sin x sin y = [cos( x − y ) − cos( x + y )] 2 1 23. cos x cos y = [cos( x − y ) + cos(x + y )] 2 1 24. sin x cos y = [sin ( x − y ) + sin ( x + y )] 2

( ( ( (

) ) ) )

⎛xm y⎞ ⎛x± y⎞ 25. sin x ± sin y = 2 cos⎜ ⎟ sin ⎜ ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎛x+ y⎞ ⎛x− y⎞ 26. cos x + cos y = 2 cos⎜ ⎟ cos⎜ ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎛x+ y⎞ ⎛x− y⎞ 27. cos x − cos y = 2 sin ⎜ ⎟ sin ⎜ ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ 28. e ± jx = cos x ± j sin x ; Euler’s Theorem

e jx + e − jx 2 jx e − e − jx 30. sin x = 2j A cos x ± B sin x = R cos( x m θ ) 31. B R = A 2 + B 2 , θ = tan −1 A A cos x ± B sin x = R sin (x ± θ ) 32. A R = A 2 + B 2 , θ = tan −1 B a b b 33. (law of sinus) = = sin A sin B sin C 34. a 2 = b 2 + c 2 − 2bc cos A (law of cosinus) 1 tan ( A − B ) a−b 2 = 35. (law of tangents) 1 a+b tan ( A + B ) 2 29. cos x =

Values of cosine, sine and exponential functions cos nπ = (− 1) sin nπ = 0 cos 2nπ = 1 sin 2nπ = 0 n nπ ⎧⎪(− 1) 2 ; n = even 5. cos =⎨ 2 ⎪⎩ 0 ; n = odd

1. 2. 3. 4.

n

n −1 nπ ⎧⎪(− 1) 2 ; n = odd =⎨ 2 ⎪⎩ 0 ; n = even n 7. e jnπ = (− 1) 8. e j 2 nπ = 1 n ⎧ nπ j ⎪ (− 1) 2 ; n = even 2 9. e =⎨ n −1 ⎪⎩ j (− 1) 2 ; n = odd

6. sin

Hyperbolic Functions 1 x (e − e − x ) 2 1 cosh x = (e x + e − x ) 2 sinh x tanh x = cosh x 1 coth x = tanh x 1 csc hx = sinh x 1 sec hx = cosh x sinh(x ± y ) = sinh x cosh y ± cosh x sinh y cosh( x ± y ) = cosh x cosh y ± sinh x sinh y

1. sinh x = 2. 3. 4. 5. 6. 7. 8.

Differentiation

1. 2. 3.

4. 5. 6. 7. 8. 9. 10. 11. 12.

d (au (x )) = a du (x ) dx dx d ( f (x ))n = n f ′(x ) ( f (x ))n −1 dx d (u (x )v(x )) = u dv(x ) + v du (x ) dx dx dx du ( x ) dv( x ) v −u d ⎛ u (x ) ⎞ dx dx ⎜ ⎟= dx ⎜⎝ v( x ) ⎟⎠ v2 d f (x) ( a ) = a f ( x ) f ′( x ) ln a dx d f (x) (e ) = e f ( x ) f ′(x ) dx d (ln f (x )) = 1 f ′(x ) dx f (x ) d (sin f (x )) = f ′(x ) cos f (x ) dx d (cos f (x )) = − f ′(x ) sin f (x ) dx d (tan f (x )) = f ′(x ) sec 2 f (x ) dx d (csc f (x )) = − f ′(x ) csc f (x ) cot f (x ) dx d (sec f (x )) = f ′(x ) sec f (x ) tan f (x ) dx

d (cot f (x )) = − f ′(x ) csc 2 f (x ) dx d f ′( x ) sin −1 f ( x ) = 14. 2 dx 1 − [ f ( x )]

13.

(

)

d f ′( x ) cos −1 f ( x ) = − 2 dx 1 − [ f ( x )] d f ′(x ) 16. tan −1 f ( x )) = ( 2 dx 1 + [ f ( x )] d f ′( x ) 17. csc −1 f ( x )) = − ( 2 dx f ( x ) [ f ( x )] − 1

(

15.

)

d f ′( x ) sec −1 f ( x ) = 2 dx f ( x ) [ f ( x )] − 1 d f ′( x ) 19. cot −1 f ( x ) = − 2 dx 1 + [ f ( x )]

18.

(

)

(

)

Indefinite Integration

1. 2.

∫ a dx = ax + c ∫ u (x ) dv = u (x )v(x ) − ∫ v(u ) du ; Integration by parts

3.

n ∫ x dx =

4.

∫x

5.

x ∫ a dx =

−1

x n +1 + c , n ≠ −1 n +1

dx = ln x + c ax +c ln a

∫ e dx = e + c 7. ∫ ln x dx = x ln x − x + c 8. ∫ sin x dx = − cos x + c 9. ∫ cos x dx = sin x + c 10. ∫ tan x dx = − ln (cos x ) + c 11. ∫ csc x dx = ln (csc x − cot x ) + c 12. ∫ sec x dx = ln (sec x + tan x ) + c 13. ∫ cot x dx = ln (sin x ) + c 14. ∫ sin x dx =x sin x + 1 − x + c 15. ∫ cos x dx =x cos x − 1 − x + c 1 16. ∫ tan x dx =x tan x − ln (1 + x ) + c 2 6.

x

x

−1

−1

−1

−1

−1

−1

2

2

2

⎛ 1 ⎞ 17. ∫ csc −1 x dx =x csc −1 x + ln⎜⎜ x + x 1 − 2 ⎟⎟ + c x ⎠ ⎝ ⎛ 1 ⎞ 18. ∫ sec −1 x dx =x sec −1 x − ln⎜⎜ x + x 1 − 2 ⎟⎟ + c x ⎠ ⎝ 1 19. ∫ cot −1 x dx =x tan −1 x + ln (1 + x 2 ) + c 2 sin(a − b) x sin(a + b) x 20. ∫ sin ax sin bx dx = − +c , a ≠b 2(a − b) 2(a + b) sin(a − b) x sin(a + b) x + +c , a ≠b 21. ∫ cos ax cos bx dx = 2(a − b) 2(a + b) cos(a − b) x cos(a + b) x − +c , a ≠b 22. ∫ sin ax cos bx dx = − 2(a − b) 2( a + b ) x sin 2ax 23. ∫ sin 2 ax dx = − +c 2 4a x sin 2ax 24. ∫ cos 2 ax dx = + +c 2 4a 25. ∫ x m sin x dx = − x m cos x + m ∫ x m −1 cos x dx

26.

∫x

m

cos x dx = x m sin x − m ∫ x m −1 sin x dx

x m e ax m m −1 ax − ∫ x e dx a a ax ae sin bx − be ax cos bx +c 28. ∫ e ax sin bx dx = a2 + b2 ae ax cos bx + be ax sin bx ax 29. ∫ e cos bx dx = +c a2 + b2 27.

m ax ∫ x e dx =

Definite Integration 2π

1.

∫ sin ax dx = 0 0

2π

2.

∫ cos ax dx = 0 0

π

3.

∫ sin

2

ax dx =

0

π

4.

∫ cos 0

2

ax dx =

π 2

π 2

⎧ 0 ; m≠n ⎪ mx nx dx = sin sin ⎨1 π ; m = n ∫0 ⎪⎩ 2 π ⎧ 0 ; m≠n ⎪ 6. ∫ cos mx cos nx dx = ⎨ 1 0 ⎪⎩ 2 π ; m = n π

5.

⎧ 0 ; m + n = even ⎪ = sin mx cos nx dx ⎨ 2m ∫0 ⎪⎩ m 2 − n 2 ; m + n = odd

π

7.

⎧π ; a>0 ⎪ ∞ ⎪ 2 sin ax 8. ∫ dx = ⎨ 0 ; a = 0 x 0 ⎪ π ; a<0 ⎪− ⎩ 2 ∞

9.

∫a 0

2

a π dx = ; a>0 2 2 +x

∞

10. ∫ e − ax sin bx dx = 0

∞

11. ∫ e − ax cos bx dx = 0

b ; a>0 a + b2 2

a ; a>0 a + b2 2

L’Hopital Rule

If f (0) = h(0) = 0 , then lim f ( x ) lim f ′( x ) = x → 0 h( x ) x → 0 h ′( x )

Limit x

lim ⎛ 1⎞ ⎜1 + ⎟ = 2.7182818284 590452354 x→∞ ⎝ x⎠ lim sin( x) 2. =1 x→0 x

1. e =

Complex Number

z = x + jy z = r/θ z = re jθ

Rectangular form Polar form Exponential form

x = r cos θ , y = r sin θ

r=

x 2 + y 2 , θ = tan −1

y x

Power Series

Taylor Series f (x + a ) = f (a ) + f ′(a )x + Maclaurin’s Series f (x ) = f (0) + f ′(0)x + Binomial Series

(1 + x )n

= 1 + nx +

f ′′(a ) 2 f ′′′(a ) 3 f n (a ) n x + x + ...... x n! 2! 3!

f ′′(0 ) 2 f ′′′(0) 3 f n (0 ) n x + x + ...... x n! 2! 3!

n(n − 1) 2 n(n − 1)(n − 2 ) 3 x + x + ..... 2! 3!

Standard Series 1 3 1 5 1 7 1 9 x + x − x + x 3! 5! 7! 9! 1 1 1 1 cos x = 1 − x 2 + x 4 − x 6 + x 8 2! 4! 6! 8! 2 3 16 5 272 7 7936 9 tan x = x + x + x + x + x 3! 5! 7! 9! 1 1 1 3 2 5 1 2 cot x = − x − x − x − x7 − x9 x 3 45 945 4725 93555 1 1 7 3 31 5 127 73 csc x = + x + x + x + x7 + x9 x 6 360 15120 604800 3421440 1 2 5 4 61 6 277 8 sec x = 1 + x + x + x + x 2 24 720 8064 1 1 1 1 1 1 1 1 ln (1 + x ) = x − x 2 + x 3 − x 4 + x 5 − x 6 + x 7 − x 8 + x 9 2 3 4 5 6 7 8 9 1 1 1 1 1 1 1 1 e x = 1 + x + x 2 + x3 + x 4 + x5 + x6 + x7 + x8 + x9 2! 3! 4! 5! 6! 7! 8! 9!

1. sin x = x − 2. 3. 4. 5. 6. 7. 8.