so do you really want to solve

1-sin3x-cosx=0 (or is it just an example)

the answer using double angle theoreom,

rational root theorem, long division in algebra, and

newton's method

gives the following answers

x = 0 + 2pik, 3pi/2 + 2pik, 0.913894794 + 2pik , and

-1.341628583 + 2pik

=== here's how

sin(3x) =sin(2x+x) = sin(2x)cos(x) + cos(2x)sin(x)

now you can use double angle identities

(2*sin(x)*cos(x)*cos(x) + (cos^2(x) -sin^2(x)) sin(x)

sin(3x) = 2*sin(x)*cos^2(x) + cos^2(x)*sin(x) - sin^3(x)

sin(3x) = 3*sin(x)cos^2(x) -sin^3(x) = 3*sin(x)*(1-sin^2(x) ) -sin^3(x) = 3*sin(x) -3*sin^3(x) -sin^3(x)

sin(3x) = 3*sin(x) -4*sin^3(x)

1-sin3x-cosx=1 -3*sin(x) +4*sin^3(x) -cos(x) = 0

isolate cos(x) term

1-3*sin(x) +4*sin^3(x) = cos(x)

square both side which could introduce extraneous answers

1 -3*sin(x) + 4*sin^3(x) -3*sin(x) +9*sin^2(x) -12*sin^4(x)

+ 4*sin^3(x) -12*sin^4(x) + 16*sin^6(x) = cos^2(x)

combine like terms

16*sin^6(x) -24*sin^4(x) + 8*sin^3(x) +9*sin^2(x) -6*sin(x) +1 = cos^2(x)

but cos^2(x) =1 -sin^2(x)

16*sin^6(x) -24*sin^4(x) + 8*sin^3(x) +9*sin^2(x) -6*sin(x) +1 = 1-sin^2(x)

16*sin^6(x) -24*sin^4(x) + 8*sin^3(x) +10*sin^2(x) -6*sin(x)=0

factor out sin(x)

sin(x) *( 16*sin^5(x) -24*sin^3 + 8*sin^2(x) +10*sin(x) - 6) = 0

so sin(x) = 0

gives the first answer of x = 0 + 2pik

checking for extraneous answer

1-sin(3*0) -cos(x) = 1 -0 -1 = 0 (check )

1- sin(3*2*pi) -cos(2pi) = 1 - 0 -1 = 0 (checks)

so 0 +2pik is one set of answers

now for the rest

16*sin^5(x) -24*sin^3 + 8*sin^2(x) +10*sin(x) - 6 = 0

divide by 2

8*sin^5(x) -12*sin^3(x) + 4*sin^2(x) + 5*sin(x) - 3 =0

let u = sin(x) /-

8u^5 -12u^3 + 4u^2 +5u -3 = 0

so first use the rational root theorem to check for rational roots

+/- 3/8 , +/- 3/4 , +/- 3/2, +/- 3/1

+/- 1/8 , +/- 1/4 , +/- 1/2 , +/- 1

if you check all the answers

x = -1 is the only root

so sin(x) = 1 gives possible roots

x = pi/2 + 2pik or 3pi/2 +2pik

so check these

checking pi/2 +2pik

1- sin(3*pi/2) -cos(pi/2) = 1 - (-1) -0 = 2 (does not check )

if you check, it does not work any of these value +2pik

so throw out all pi/2 +2pik answers

checking 3*pi/2 +2pik

1 - sin(3*(3*pi/2) -cos(3pi/2) = 1- sin(9pi/2) -0 =1 -1 -0=0 (checks)

It also check of the 2pik if you check

so far we have answers

0 + 2pik

3pi/2 + 2pik

so we had

8u^5 -12u^3 + 4u^2 +5u -3 = 0

but u=-1 was a root

so (u+1) is a factor

(8u^5 -12u^3 + 4u^2 +5u -3 )/ (u+1) = 8u^4 -8u^3 -4u^2 +8u -3

Now the bad news

y=8x^4 -8x^3 -4x^2 +8x -3

If you use the rational root theorem

+/- 3/8 , +/- 3/4 , +/- 3/2, +/- 3/1

+/- 1/8 , +/- 1/4 , +/- 1/2 , +/- 1

but none of these works

so to find approximate answer , use newton's method to get estimates

if you graph "8x^4 -8x^3 -4x^2 +8x -3" , you can see it show two roots.

One root is between 0 and 1 and one root between -1 and 0

so take 0.5 as the first guess between 0 and 1

f(x) = 8x^4 -8x^3 -4x^2 +8x -3

f'(x) = 32x^3 -24x^2 -8x + 8

x_previous_guess = x - (8x^4 -8x^3 -4x^2 +8x -3)/ (32x^3 -24x^2 -8x + 8 )

Using newton method,

it gives the following 0.791888159

so we have possible answer from

sin(x) = 0.791888159

x = asin(0.913894794)

this gives

x = 0.913894794 + 2pik

x = (pi- 0.913894794) +2pik = 2.227697859 + 2pik

so check these answer for extraneous answer

1- sin(3* 0.913894794 ) -cos(0.913894794) = closest enough to 0

1 -sin(2.227687859*3) -cos( 2.227697859 ) = 1.22 (does not check )

so far with have

0 + 2pik , 3pi/2 +2pik , 0.913894794 + 2pik

One more root to check , the one between -1 and 0 ,

starting with -1/2 as the root using this same equation

x_previous_guess = x - (8x^4 -8x^3 -4x^2 +8x -3)/ (32x^3 -24x^2 -8x + 8 )

(-1/2) gives us the same root as before, so we need to use a different first guess

The graph of 8x^4 -8x^3 -4x^2 +8x -3 shown the second root is closer to -1 than 0 ,

so let's try -0.75 as the 1st guess

this gives the 2nd root as -0.973855793

since u= sin(x)

x= asin( -0.973855793)

x = -1.341628583 + 2pik or

pi- (-1.341628583) +2pik = 4.483221236+ 2pik

again check for extraneous answer

1- sin(3* -1.341628583) -cos(-1.341628583) = 0 (checks within rounding error)

so one more answer is

x= -1.341628583 + 2pik

1 -sin(4.483221236*3) -cos( 4.483221236 ) = 0.445 (does not checks)

so through out , x = 4.483221236+ 2pik

so one more answer is

x= -1.341628583 + 2pik

so we now have all the answer (I think )

x = 0 + 2pik , 3pi/2 + 2pik , 0.913894794 + 2pik , and -1.341628583 + 2pik