well frnz , here it goes , some useful shortcuts in trigo etc....

it will help u in cbse also for checking ur answer....

HOPE U LIKE IT


1) The value of

sin(pi/14).sin(3pi/14).sin(5pi/14).sin(7pi/14).sin(9pi/14).sin(11pi/14).sin(13pi/14)


here's the shortcut

value = 1 / 2ⁿ -1

n = no. of terms

so ans = 1/64


condition (standardization shud be there).........(denomenator shud be even)


2)The value of cos(2pi/15).cos(4pi/15).cos(8pi/15)cos(14pi/15)

shortcut

ans = 1/2ⁿ

ans = 1/16

condition (den. must be odd)


3)The angle fr cos@ = tan@

is 2 sin 18 degrees


4)Sin(pi/7) + sin(2pi/7) + sin(3pi/7)


shortcut

value = 1/2 cot (1/2 (first or smallest angle))

= 1/2 cot ( (1/2).(pi/7))

= 1/2 cot (pi/14)


condition ----angle must be in series.... (pi 2pi 3pi etc)


5)and in sin qs if denomenator is odd

sin (pi/7).sin(2pi/7).sin(3pi/7)

shortcut

 = (denomenator)^1/2 / 2ⁿ

= root7 / 8

its the ans



Functions

f(x) = x³ + 5

then f^-1(x) = ????????

simple yaar

y = x³ + 5

y - 5 = x³

now just interchange x and y

x - 5 = y³........................this is the major step
i.e (x - 5)^1/3 = y



matrices

trace of a matrix ---add the elements of primary diagonals....

1.....-5.....7
0.....7......9
11....8.....9

trace = 1 + 7 + 9 = 17
its the ans

hope these helps



cheers!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!