Question

17. solve the equation for exact solutions over the interval (0.211). 2 cos²x+ 5 cosx-3=0

Answer 1

Given equation is 2cos²x + 5cosx - 3 = 0

Let cosx = p , then we get

2p² + 5p - 3 = 0

Solve this by splitting the middle term method,

2p² + 5p - 3 = 0

2p² + 6p - p - 3 = 0

2p(p + 3) - 1(p + 3) = 0

(2p - 1)(p + 3) = 0

(2p - 1) = 0   or     (p + 3) = 0

p = (1/2)    or      p = -3

cosx = (1/2)     or     cosx = (-3)

cosine has a range [-1,1], hence cosx = (-3) is not possible.

Hence we get,

cosx = (1/2)             [cosine is positive in first and fourth quadrant]

x = cos⁻¹(1/2)

x = (π/3) and    x = (2π - (π/3)) = (5π/3)

Hence x = (π/3) , (5π/3) are the required solutions.