Question

1. an angle in standard position has a measure of 200°. identify a corresponding coterminal angle.

a. \\( \frac{\pi}{9} \\)
b. \\( \frac{10\pi}{9} \\)
c. \\( \frac{-7\pi}{9} \\)
d. \\( \frac{17\pi}{9} \\)
e. \\( \frac{8\pi}{9} \\)

1. which of the following is a solution to the equation \\( x^{4}-27 x^{2}-14 x + 120 = 0 \\)?

a. 1
b. -2
c. 3
d. -4
e. -5

1. which of the following is not a solution to \\( t^{4}+6 t^{2}-27 = 0 \\)?

a. \\( t = 3 \\)
b. \\( t = \sqrt{3} \\)
c. \\( t = -\sqrt{3} \\)
d. \\( t = 3i \\)
e. \\( t = -3i \\)

1. find all solutions to the equation \\( x^{4}+11 x^{2}+30 = 0 \\).

a. \\( x = \pm 5, x = \pm 6 \\)
b. \\( x = \pm 5i, x = \pm 6i \\)
c. \\( x = \pm i \sqrt{5}, x = \pm i \sqrt{6} \\)
d. \\( x = \pm \sqrt4{5}, x = \pm \sqrt4{6} \\)
e. \\( x = \pm i \sqrt4{5}, x = \pm i \sqrt4{6} \\)

Explanation:

Step1: Convert degree to radian

We know that \(1^{\circ}=\frac{\pi}{180}\) radians. So \(200^{\circ}=200\times\frac{\pi}{180}=\frac{10\pi}{9}\) radians.

Step2: Check coterminal angles

Coterminal angles differ by \(2k\pi\) (\(k\in\mathbb{Z}\)). For \(k = 0\), the angle \(\frac{10\pi}{9}\) is coterminal with itself.

Step1: Substitute \(x=-4\) into the equation \(x^{4}-27x^{2}-14x + 120\)

$$
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$$

Step1: Substitute \(t = 3\) into the equation \(t^{4}+6t^{2}-27\)

$$
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$$

Step1: Let \(y=x^{2}\), then the equation \(x^{4}+11x^{2}+30 = 0\) becomes \(y^{2}+11y + 30=0\)

Factor the quadratic equation: \(y^{2}+11y + 30=(y + 5)(y+6)=0\)
So \(y=-5\) or \(y=-6\)

Step2: Substitute back \(y = x^{2}\)

If \(x^{2}=-5\), then \(x=\pm i\sqrt{5}\); if \(x^{2}=-6\), then \(x=\pm i\sqrt{6}\)

Answer:

B. \(\frac{10\pi}{9}\)