Question

Решите неравенство: cos2x-cos^2x-2cosx<-2

Answer 1

Решение
cos2x-cos^2x-2cosx<-2
2cos²x - 1 - cos²x - 2cosx < - 2
cos²x - 2cosx + 1 < 0
cosx = t
t² - 2t + 1 = 0
(t - 1)² = 0
t = 1
cosx < 1
2πn < x < 2π + 2πn, n ∈ Z

Answer 2

I can help you with math problems. To solve this inequality, we need to use some trigonometric identities and algebraic manipulations. Here are the steps:

- Rewrite cos2x as 2cos^2x - 1 using the identity . - Move all the terms to one side and factor out cosx: cosx(2cosx - 3)(cosx + 1) < -1 - Find the zeros of the polynomial: cosx = 0, cosx = 3/2, cosx = -1 - Since cosx = 3/2 is impossible, we only consider the other two zeros. - Draw a sign chart and test the intervals between the zeros: (-pi/2, 0), (0, pi/2), (pi/2, pi), (pi, 3pi/2), (3pi/2, 2pi) - The solution set is the union of the intervals where the polynomial is negative: (-pi/2, 0) U (pi, 3pi/2) - Divide both sides by 2 to get the solution for x: (-pi/4, 0) U (pi/2, 3pi/4)

You can check your answer using a graphing calculator or an online solver . I hope this helps. Have a nice day.