Question

Solve `sin^2(x)=cos(x` )for all solutions 0≤x<2π. Ask for X? Give your answers accurate to 2 decimal places, as a list separated by commas

Answer 1

`51^@83, 308^@17`

Explanation:

`sin^2 x = cos x`
`(1 - cos^2 x) = cos x`
`cos^2 x + cos x - 1 = 0`.
Solve this quadratic equation for cos x.
`D = d^2 = b^2 - 4ac = 1 + 4 = 5` --> `d = +- sqrt5`
There are 2 real roots:
`cos x = -b/(2a) +- d/(2a) = - 1/2 +- sqrt5/2 = (-1 +- sqrt5)/2`
a. `cos x = (-1 - sqrt5)/2 = -3.24/2 = -1.62` (rejected as < - 1)
b. `cos x = (-1 + sqrt5)/2 = 0.62`
Calculator and unit circle give 2 solutions for x -->
`x = +- 51^@83`
Note. x = - 51.83 is co-terminal to `x = 360 - 51.83 = 308^@17`
Answers for `(0, 2pi)`
`51^@83, 308^@17`
Check by calculator.
x = 308.17 --> `sin^2 x = 0.62` --> cos x = 0.62. Proved