precalc
posted by bandit .
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.
Respond to this Question
Similar Questions

calc
What are the intercepting points of y^2=4x and x^2=y I set both equations equal to 0, but only get x=0 and there is another intercepting point. please help! You can solve for one of the equation and substitute it into the other equation. … 
confused calc
dy/dx=2y^2 and if y=1 when x=1, then when x=2, y=? 
PreCalc
I need to figure out how to solve the equation xsquared ysquared+xy=1 i have no clue. I would appreciate any help! did you mean (x^2)(y^2) + xy = 1 ? 
PreCalc
To solve an equation by finding xintercepts, Cynthia graphed the function y=x^35x+2x3. Give several possibilities for the original equation. 
PreCalc
Solve the equation. (tan^2)3x=3 
Trig/PreCalc
Could someone please help me on this problem. I'm really confused, and this assignment is due tomorrow. Solve the logarithmic equation algebraically. Approximate the result to three decimal places. log4xlog4(x1)=1/2 To start off … 
PreCalc
How do I solve this equation? x+5 = sqrt(x+5) 
PreCalculus
Solve the equation 2x + 5y  3z = 1. Write the general solution as a matrix equation. [x] [ ] [1] [y]= [ ] X [s] [z] [ ] [t] I'm supposed to fill in that blank in the middle, but I don't understand how to solve this if I'm only given … 
PreCalc
How do I solve this equation? 4^x * (1/2)^32x = 8*(2^x)^2 
PreCalc
Solve the quadratic equation by factoring. Check your solutions in the original equation. (Enter your answers as a commaseparated list.) x^2 + 2ax + a^2 = 0