precalc
posted by bandit on .
how do i solve this equation algebraicly
solve the equation 1+25cos(x)=25sin^2(x)

Substitute sin²(x)=1cos²(x).
This results in a quadratic equation in cos(x). Solve for y=cos(x) and evaluate x=arccos(y) if 1≤y≤1. 
1+25cos(x)=25sin^2(x)
25 sin^2 x + 25 cos x + 1
25 (1  cos^2 x) + 25 cos x + 1
25 + 25 cos^2 x + 25 cos x + 1
25 cos^2 x + 25 cos x  24
substitute u = cos x
25u^2 + 25u  24
complete the square
u^2 + u  24/25 = 0
u^2 + u = 24/25
u^2 + u + 1/4 = 121/100
(u + 1/2)^2 = 121/100
take square root of both sides
 u + 1/2  = 11/10
u + 1/2 = 11/10 and u + 1/2 = 11/10
u = 8/5 and u = 3/5
substitute back in u = cos x
cos x = 8/5 and cos x = 3/5
x = arccos (8/5)
x = arccos (3/5) 
forgot to put
I am not a tutor
I think this is right 
It's almost perfect.
In the last step,
"x = arccos (8/5)
x = arccos (3/5) "
we have to reject the first value of arccos(8/5) because arccos(8/5) does not exist.
For a solution between 0 and 2π, there are two values of x that satisfy arccos(3/5), i.e.
approx. 53 degrees and 36053 degrees.
For the general solution, add 2kπ to the solutions, where k is an integer.