To solve for x in the given equation 2 + 2cosx = 3sinx * cosx + 2sinx, we need to rearrange terms and simplify.
2 + 2cosx = 3sinx * cosx + 2sinx=> 2 + 2cosx = sinx (3cosx + 2)
Now, let's use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to substitute for sin^2(x) in terms of cosx:
sin^2(x) = 1 - cos^2(x)
Substitute this into the equation:
2 + 2cosx = sinx (3cosx + 2)=> 2 + 2cosx = sinx 3cosx + sinx 2=> 2 + 2cosx = 3sinxcosx + 2sinx=> 2 + 2cosx = 2sinx + 3sinxcosx
Next, use sin^2(x) + cos^2(x) = 1:
2 + 2cosx = 2sinx + 3sinxcosx=> 2 + 2cosx = 2sinx + 3sinxcosx=> 2 + 2cosx = 2sinx + 3sinxcosx=> 2 + 2cosx = 2sinx + sinx(3cosx)=> 2 + 2cosx = 2sinx + sinx(3cosx)=> 2 + 2cosx = sinx(2 + 3cosx)
Now we have:
2 + 2cosx = sinx(2 + 3cosx)
To continue solving for x, we can square both sides to eliminate the square root:
(2 + 2cosx)^2 = sinx^2(2 + 3cosx)^24 + 8cosx + 4cosx^2 = sinx^2(4 + 12cosx + 9cosx^2)4 + 8cosx + 4cosx^2 = sinx^2(4 + 12cosx + 9cosx^2)
Further simplification may involve expanding and manipulating the resulting equation algebraically to isolate the variable x. The final solution for x will be the values that satisfy the equation.
To solve for x in the given equation 2 + 2cosx = 3sinx * cosx + 2sinx, we need to rearrange terms and simplify.
2 + 2cosx = 3sinx * cosx + 2sinx
=> 2 + 2cosx = sinx (3cosx + 2)
Now, let's use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to substitute for sin^2(x) in terms of cosx:
sin^2(x) = 1 - cos^2(x)
Substitute this into the equation:
2 + 2cosx = sinx (3cosx + 2)
=> 2 + 2cosx = sinx 3cosx + sinx 2
=> 2 + 2cosx = 3sinxcosx + 2sinx
=> 2 + 2cosx = 2sinx + 3sinxcosx
Next, use sin^2(x) + cos^2(x) = 1:
2 + 2cosx = 2sinx + 3sinxcosx
=> 2 + 2cosx = 2sinx + 3sinxcosx
=> 2 + 2cosx = 2sinx + 3sinxcosx
=> 2 + 2cosx = 2sinx + sinx(3cosx)
=> 2 + 2cosx = 2sinx + sinx(3cosx)
=> 2 + 2cosx = sinx(2 + 3cosx)
Now we have:
2 + 2cosx = sinx(2 + 3cosx)
To continue solving for x, we can square both sides to eliminate the square root:
(2 + 2cosx)^2 = sinx^2(2 + 3cosx)^2
4 + 8cosx + 4cosx^2 = sinx^2(4 + 12cosx + 9cosx^2)
4 + 8cosx + 4cosx^2 = sinx^2(4 + 12cosx + 9cosx^2)
Further simplification may involve expanding and manipulating the resulting equation algebraically to isolate the variable x. The final solution for x will be the values that satisfy the equation.