Question

Solve the following equation cot x-(square root of 5)cosec x=2 Find the maximum and minimum for
cos x-2sinx+3 .

Answer 1

cot x - sqrt5 csc x =2

(cos x - sqrt5)/sin x = 2

cos x - sqrt5 = 2sin x

square both sides
cos^2 -2sqrt5*cos x +5 = 4sin^2 = 4(1-cos^2)

5cos^2 -2sqrt5*cos x +1 = 0

(sqrt5cos(x) -1)^2 = 0

sqrt5*cos x = 1

cos x = 1/sqrt5

x = cos^-1 (1/sqrt5) = 63.43 degrees

Answer 2

oops, 63.43 degrees doesn't work
the other possible case is 360 - 63.43 = 296.57 degrees
this way cos still equals 1/sqrt5

Answer 3

to find max/min, differentiate and set equal to zero

f'(x) = -sin x -2cos x = 0
-2cos x = sin x
tan x = -2
x = tan^-1 (-2)
x = 116.57 and 296.57

min = 116.57
max = 296.57