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ok,
7(i) first expand 'cos(x+45)'
cos(x+45) = cosxcos45 - sinxsin45 [<---hope you know the addition formula for cos(A +B)]
=(√2 /2)cosx - (√2 /2)sinx
so now we have to express "cos(x + 45) - (√2)sinx " in the form "Rcos(x + a)" [cos(x + 45) - (√2)sinx = Rcos(x + a)]
first i'll simplify the LHS (the one in blue):
cos(x + 45) - (√2)sinx = (√2 /2)cosx - (√2 /2)sinx - (√2)sinx
= (√2 /2)cosx - (3√2 /2)sinx
now i'm expanding the RHS(the one in green):
Rcos(x + a) = Rcosxcosa - Rsinxsina
i'll equate the RHS and the LHS :
Rcosxcosa - Rsinxsina = (√2 /2)cosx - (3√2 /2)sinx <---i can re-write this as:
(Rcosa)cosx - (Rsina)sinx = (√2 /2)cosx - (3√2 /2)sinx
now to find R:
R = √[(√2 /2)^2 + (-√3 /2)^2]
= 2.236
now to find the angle 'a' :
(Rcosa)cosx - (Rsina)sinx = (√2 /2)cosx - (3√2 /2)sinx
from the equation above^, notice the coefficients of 'cosx' and 'sinx' respectively
so ===> Rcosa = √2 /2 and Rsina= 3√2 /2
to find the angle, we need to use 'tan'
hence; tana = [3√2 /2] / [√2 /2] because tan = sin/cos
angle 'a' = 71.57
so u'll have: 2.236cos(x + 71.57) nd u've expressed it in the form they want
(btw the working is not this long)
7(ii) now we solve: cos(x + 45) - (√2)sinx = 2 [the RHS is equivalent to the answer we found in (i)]
so,
2.236cos(x + 71.57) = 2
cos (x + 71.57) = 0.8945 [to make my working easier, i let 'x + 71.57" = y]
==> cosy = 0.8945
y = 26.56
cos is positive in the first and fourth quadrant so,
y = 360 - 26.56 = 333.4
nd i also added 360 to 26.56 because the range of 'x' changed i.e 26.56 < x + 26.56 < 386.56
so another value of y= 386.56
now we've got y as 26.56, 333.4 and 386.56
so frm these we can find our 'x' where "x = y - 71.57"
===> x = 261.8 and 315.0
hope i explained properly nd didn't confuse u evn more
thnks.. i forgot to put the sin 45 and cos 45 to (√2 /2)