91Ó°ÊÓ

State the quadrant of each complex number, then write it in trigonometric form.Answer in degrees. $$-2-2 i$$

Use De Moivre's theorem to verify the solution given for each polynomial equation. $$2 z^{4}+3 z^{3}-4 z^{2}+2 z+12=0 ; z=1-i$$

Solve using the law of sines and a scaled drawing. If two triangles exist, solve both completely. $$\begin{aligned} &\text { side } c=25.8 \mathrm{mi}\\\ &\angle A=30^{\circ}\\\ &\text { side } a=12.9 \mathrm{mi} \end{aligned}$$

Solve using the law of sines and a scaled drawing. If two triangles exist, solve both completely. $$\begin{aligned} &\begin{aligned} \text { side } c &=58 \mathrm{mi} \\ \angle C &=59^{\circ} \end{aligned}\\\ &\text { side } b=67 \mathrm{mi} \end{aligned}$$

For each pair of vectors given, (a) compute the dot product \(\mathbf{p} \cdot \mathbf{q}\) and \((\mathrm{b})\) find the angle between the vectors to the nearest tenth of a degree. $$\mathbf{p}=\langle 5,2\rangle ; \mathbf{q}=\langle 3,7\rangle$$

Equilateral triangles in the complex plane: \(u^{2}+v^{2}+w^{2}=u v+u w+v w\) If the line segments connecting the complex numbers \(u, v,\) and \(w\) form the vertices of an equilateral triangle, the formula shown above holds true. Verify that \(u=2+\sqrt{3} i, v=10+\sqrt{3} i,\) and \(w=6+5 \sqrt{3} i\) form the vertices of an equilateral triangle using the distance formula, then verify the formula given.

Electric current: In the United States, electric power is supplied to homes and offices via a " 120 V circuit," using an alternating current that varies from \(170 \mathrm{V}\) to \(-170 \mathrm{V},\) at a frequency of 60 cycles/sec. (a) Write the voltage equation for U.S. households, (b) create a table of values illustrating the voltage produced every thousandth of a second for the first half-cycle, and (c) find the first time \(t\) in this half-cycle when exactly \(140 \mathrm{V}\) is being produced.

Determine if the pair of vectors given are orthogonal. $$\mathbf{u}=\langle-3.5,2.1\rangle ; \mathbf{v}=\langle-6,-10\rangle$$

Find comp_u for the vectors \(\mathbf{u}\) and \(\mathbf{v}\) given $$\mathbf{u}=\langle 3,5\rangle ; \mathbf{v}=\langle-7,1\rangle$$

Approaching from the west, a group of hikers notes the angle of elevation to the summit of a steep mountain is \(35^{\circ}\) at a distance of 1250 meters. Arriving at the base of the mountain, they estimate this side of the mountain has an average slope of \(48^{\circ} .\) (a) Find the slant height of the mountain's west side. (b) Find the slant height of the east side of the mountain, if the east side has an average slope of \(65^{\circ} .\) (c) How tall is the mountain? (IMAGE CANNOT COPY)