Question

Question #e0bdc

Answer 1

`x = +- 41^@41`
`x = +- 60^@`

Explanation:

`8 - 7cos x = 6sin^2 x`
Replace `sin^2 x` by (`1 - cos^2 x`)
`8 - 7cos x == 6 - 6cos^2 x`
`6cos^2 x - 7cos x + 2 = 0`
Solve this quadratic equation for cos x by using the improved quadratic formula (Socratic, Google Search)
`D = d^2 = b^2 - 4ac = 49 - 48 = 1` --> `d = +- 1`
There are 2 real roots:
`cos x = -b/(2a) +- d/(2a) = 7/12 +- 1/12 = (7 +- 1)/12`
`cos x = 8/12 = 0.75`, and
`cos x = 6/12 = 1/2`

a. cos x = 0.75
Calculator and unit circle give 2 solutions:
`x = +- 41^@41`
b. `cos x = 1/2` --> 2 solutions:
`x = +- 60^@`