Find, to the nearest hundredth place, the solutions to the following equation(s), where 0 ≤ x ≤ 360 degrees. `(sec(x)-1)(2sec(x)+3)=3` Thank you to everyone that helps.
`(secx -1 )(2sec x + 3 ) = 3`
To solve this, first, multiply the right side. To do so, apply FOIL.
`2sec^2x +3secx - 2secx - 3 = 3`
`2sec^2x + secx - 3 = 3`
Then, set one side equal to zero. So, subtract both sides by 3.
`2sec^2x + secx - 3 - 3 =3 - 3`
`2sec^2x + secx - 6 =0`
Then, factor the left side.
`(2sec x -3)(secx + 2) = 0`
Set each factor equal to zero and solve for x.
For the first factor:
`2secx - 3 = 0`
`2secx = 3`
`secx = 3/2`
`cosx = 2/3`
Since cosine is positive, the other value of the angle x lies on the fourth quadrant. Hence, the other angle is:
`x = 311.81^o`
For the second factor:
`secx + 2 = 0`
`secx = -2`
`cosx = -1/2`
Since cosine is negative, the other angle x lies at the third quadrant. So, the other angle is:
`x = 240^o`
Therefore, the solution to the given equation in the given interval are `x=48.19^o, 120^o, 240^o,` and `311.81^o` .