Final answer:
To find the value of cos 2x and the common ratio of the geometric sequence, trigonometric identities such as the double-angle identities for sine and cosine are used. The common ratio is cos 2x. Further information about x is needed to find an explicit numerical value.
Step-by-step explanation:
The question involves finding the numerical value of cos 2x given that sin x, sin2 2x, and cos x · sin 4x form an increasing geometric sequence. We will make use of trigonometric identities to find the answer.
To find the numerical value of cos 2x, and the common ratio of the geometric sequence, we recognize that:
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- sin 2x = 2 sin x cos x (Double-angle identity for sine)
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- sin 4x = 2 sin 2x cos 2x (Double-angle identity for sine, applied twice)
Thus, sin 4x = 2 sin 2x cos 2x = 4 sin x cos x cos 2x. If we divide cos x · sin 4x by sin2 2x, we get:
cos x ((4 sin x cos x cos 2x)/(4 sin2 x cos2 x)) = cos 2x.
Since the sequence is increasing and geometric, the common ratio is cos 2x, which is positive.
To find an explicit value for cos 2x, further information about the angle x or the terms of the geometric sequence is needed, as certain values of x will give different outcomes for cos 2x.
However, if we apply the identity cos 2x = 1 - 2 sin2 x, we could potentially solve for x given more information, and thus find cos 2x explicitly.