Number of roots of trigonometric equation

Geometry Level 3

In the domain 0 < x < 4 π , 0<x<4\pi, how many roots does the following equation have: 3 sin ⁡ 2 x + 4 cos ⁡ 2 x 2 − 4 = 0 ? 3\sin^2x+4\cos^2\frac{x}{2} -4 =0?


The answer is 5.

Sandeep Mehra by Sandeep Mehra , 23, India

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1 solution

Arko Bhaumik
Jun 1, 2014

First simplify the given equation by eliminating the only submultiple. Express the latter in terms of x. Simplify the expression and express (sinx)^2 as 1-(cosx)^2. The values of cosx come out as either 1 or -0.33. Thus the general solutions are x=2n(pi), and x=2n(pi)+(109.47) or 2n(pi)-(109.47). Here, 109.47 is in degrees, it is the inverse circular function of cosx= -0.33. Bounded between 0 and 4(pi), there are five permissible solutions in total for the two cases.

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