Help! Im stuck! 4sin^2x + 2cos^2x = 3, 2cosx - 3tanx = 0

[stellarluv]

OK.. I really dont know where to begin... 4sin^2x + 2cos^2x = 3, 2cosx - 3tanx = 0
any help would be greatly appreciated. Thanks. :?

 

[skeeter]

4sin²x + 2cos²x = 3

2sin²x + 2sin²x + 2cos²x = 3

2sin²x + 2(sin²x + cos²x) = 3

2sin²x + 2 = 3

2sin²x = 1

sin²x = 1/2

sinx = +/- sqrt(1/2) = +/- sqrt(2)/2

x = pi/4, 3pi/4, 5pi/5, and 7pi/4


2cosx - 3tanx = 0

2cosx - 3sinx/cosx = 0

[2cos² - 3sinx]/cosx = 0

set the numerator = 0 ...

2(1 - sin²x) - 3sinx = 0

2 - 2sin²x - 3sinx = 0

0 = 2sin²x + 3sinx - 2

0 = (2sinx - 1)(sinx + 2)

sinx = 1/2 , x = pi/6, 5pi/6

sinx = -2 has no solution

 

[stellarluv]

Very very helpful! Thank you!