Question

the following equation involves a trigonometric equation in quadratic form
2 sin²x = 5 sinx + 7
select the correct choice below and, if necessary, fill in the an
o a. x=
(type an exact answer in terms of π. use integ
o b. there is no solution.

Explanation:

Step1: Rewrite the equation

$$2\sin^{2}x - 5\sin x - 7 = 0$$
Let \(t=\sin x\), then the equation becomes \(2t^{2}-5t - 7 = 0\)

Step2: Solve the quadratic equation

Use the quadratic formula \(t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) for \(at^{2}+bt + c = 0\). Here \(a = 2\), \(b=-5\), \(c=-7\)

$$
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$$

We get \(t_{1}=\frac{5 + 9}{4}=\frac{14}{4}=\frac{7}{2}\) and \(t_{2}=\frac{5-9}{4}=-1\)

Step3: Substitute back \(t = \sin x\)

Since \(-1\leqslant\sin x\leqslant1\), for \(t=\frac{7}{2}\), \(\sin x=\frac{7}{2}\) has no solution.
For \(t=-1\), \(\sin x=-1\), then \(x=\frac{3\pi}{2}+2k\pi,k\in\mathbb{Z}\)

Answer:

A. \(x=\frac{3\pi}{2}+2k\pi,k\in\mathbb{Z}\)