the following equation involves a trigonometric equation in quadratic form. solve the equation on the…

the following equation involves a trigonometric equation in quadratic form. solve the equation on the interval 0,2π).\n3 sin²x = 7 sinx - 4\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\noa. x=\n(type an exact answer in terms of π. use integers or fractions for any numbers in the expression. use a comma\nto separate answers as needed.)\nob. there is no solution.

the following equation involves a trigonometric equation in quadratic form. solve the equation on the interval 0,2π).\n3 sin²x = 7 sinx - 4\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\noa. x=\n(type an exact answer in terms of π. use integers or fractions for any numbers in the expression. use a comma\nto separate answers as needed.)\nob. there is no solution.

Answer

Explanation:

Step1: Rearrange the equation

$$3\sin^{2}x - 7\sin x + 4=0$$ Let (t = \sin x), then the equation becomes (3t^{2}-7t + 4 = 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 = 3), (b=-7), (c = 4). First, calculate the discriminant (\Delta=b^{2}-4ac=(-7)^{2}-4\times3\times4=49 - 48=1). Then (t=\frac{7\pm\sqrt{1}}{6}=\frac{7\pm1}{6}). We get (t_{1}=\frac{7 + 1}{6}=\frac{8}{6}=\frac{4}{3}), (t_{2}=\frac{7-1}{6}=1).

Step3: Substitute back (t=\sin x)

Since (- 1\leqslant\sin x\leqslant1), for (t=\frac{4}{3}), (\sin x=\frac{4}{3}) has no solution (because (\frac{4}{3}>1)). For (t = 1), (\sin x=1). On the interval ([0,2\pi)), (x=\frac{\pi}{2}).

Answer:

A. (x=\frac{\pi}{2})