Solve the equation : sin x - 2 cos x = 2.?

Answer 1

Reducing the original equation to the half angle formula of tangents, it is reduced to :

[2 tan (x/2) /1 + tan (x/2)^2] - 2[{1- tan (x/2)^2}/{1 + tan (x/2)^2}= 2
whence tan (x/2) = 2 and x = 2arc tan2+ 2n.pi,
n =......-2,-1,0,1,2.........

However this formula doesn't include all the solutions: as can be readily verified, all the angles x = (2n + 1)pi, n=....-2,-1,0,1,2.....are also the solutions. These angles were lost when tangent of half-angle was introduced. The original equation was meaningful for all the values of x while the second equation was meaningful only when tan(x/2) is meaningful, that is, for x not equal to (2n+1)pi.

Answer 2

The max and min of the sin and cos functions are 1 or -1.

When sin x = 0, cos x = +1 or -1.

This happens when x=0 (cos x = 1)
or x=180 (or pi radians) (cos x =-1)

substituting 0 for sin x, then -2 cos x = -2(1) = -2
or -2 cos x = -2(-1) = 2.

Therefore, x = 180 degrees or pi radians.

Answer 3

sin x - 2 cos x = 2
tan x - 2 = 2 sec x + 2
tan2 x= 4 sec2 x + 8 sec x + 4 (square both sides)
sec2 x - 1 = 4 sec2 x + 8 sec x + 4
3 sec2 x + 8 sec x + 5= 0

let y=sec x

3 y^2 + 8y + 5 = 0
(solve this quadratic equation for values of y, and substitute to original equation y= sec x to get values of x)

you'll get y=-1 as the only valid answer.
sec x= -1
1/cos x= -1
cos x= -1

refer to graph of y=cos x

x= 180 degrees or pi radians

Answer 4

The obvious answer is when x=180°, but there is also a solution when sin(x)=.8 and cos(x)=-.6 when x=127.87°

Answer 5

x = 180 degrees = pi

Answer 6

sinx-2cosx=2
sinx-2(1-sin^2x)^1/2=2 transposing
sinx-2=2(1-sin^2x)^1/2 squaring both sides
sin^2x-4sinx+4=4-4sin^2x
=>5sin^2x-4sinx=0
=>sinx(5sinx-4)=0
=>sinx=0 or sinx=4/5