FM Pure U4

Question 1

sin(2a)

Answer 1

2sin(a)cos(a)

Question 2

cos(2a)

Answer 2

cos^2(a) - sin^2(a)

Question 3

tan(2a)

Answer 3

2tan(a) / 1-tan^2(a)

Question 4

General solutions sin=

Answer 4

n(pi) + (-1)^nPV

Question 5

General solutions cos=

Answer 5

2n(pi) +- PV

Question 6

General solutions tan=

Answer 6

n(pi) + PV

Question 7

How to solve asin(x) + bcos(x)

Answer 7

=rsin(x)cos(c) + rcos(x)sin(c)
Where a=rcos(c) and b=rsin(c)
r= sqrt(a^2 + b^2) and c - tan^(-1)(b/a)

Question 8

Small angles

Answer 8

sin(dx) ~ tan(dx) ~ dx
cos(dx) ~ 1- (1/2)(dx)^2

Question 9

arg(a + ib)

Answer 9

tan^(-1) (b/a)

Question 10

De Moivre a+ib =

Answer 10

= r(cos(c) + isin(c))

Question 11

If z^n = cos(nc) + isin(nc),
z^n + z^-n =
z^n - z^-n =

Answer 11

z^n + z^-n = 2cos(nc)
z^n - z^-n = 2isin(nc)

Question 12

If z^n = cos(nc) + isin(nc),
2cos(nc) =
2isin(nc) =

Answer 12

2cos(nc) = z^n + z^-n
2isin(nc) = z^n - z^-n

Question 13

Complex roots of unity often in form. Sum of roots

Answer 13

1, w, w^2. Sum = 0

Question 14

Z^n = 1 (roots of unity) is given by

Answer 14

formula book

Question 15

int(1/x)

Answer 15

ln(x) + c

Question 16

tan^2 + 1 =

Answer 16

sec^2

Question 17

1 + cot^2 =

Answer 17

cosec^2

Question 18

sec^2 in terms of tan

Answer 18

tan^2 + 1

Question 19

cosec^2 in terms of cot

Answer 19

1 + cot^2

Question 20

Vol of revolution abt x:

Answer 20

pi int(y^2)dx

Question 21

Vol of revolution abt y

Answer 21

pi int(x^2)dy

Question 22

cosh(x)

Answer 22

1/2(e^x + e^-x)

Question 23

sinh(x)

Answer 23

1/2(e^x - e^-x)

Question 24

Converting trig to hyperbolics, when does sign change?

Answer 24

On product of sines.

Question 25

How would you deal with int(sqrt(a^2 - x^2))?

Answer 25

Substitute x=asinu

Question 26

How would you deal with int(a^2 + x^2)?

Answer 26

Substitute x=atanu (or partial fractions, I suppose)

Question 27

How would you deal with int(sqrt(x^2 - a^2))?

Answer 27

Substitute x=acoshu

Question 28

How would you deal with int(sqrt(x^2 + a^2))?

Answer 28

Substitute x=asinhu