91Ó°ÊÓ

Concept Check The complex number \(z,\) where \(z=x+y i,\) can be graphed in the plane as \((x, y) .\) Describe the graphs of all complex numbers z satisfying the conditions. The imaginary part of \(z\) is 1

suppose \(z=r(\cos \theta+i \sin \theta)\) Use vectors to show that $$ -z=r[\cos (\theta+\pi)+i \sin (\theta+\pi)] $$

A flagpole 95.0 ft tall is on the top of a building. From a point on level ground, the angle of elevation of the top of the flagpole is \(35.0^{\circ},\) and the angle of elevation of the bottom of the flagpole is \(26.0^{\circ} .\) Find the height of the building.

Find the area of the Bermuda Triangle if the sides of the triangle have approximate lengths \(850 \mathrm{mi}, 925 \mathrm{mi}\), and \(1300 \mathrm{mi}\).

The graph of \(r=a \theta\) in polar coordinates is an example of the spiral of Archimedes. With your calculator set to radian mode, use the given value of a and interval of \(\theta\) to graph the spiral in the window specified. $$a=1,0 \leq \theta \leq 4 \pi,[-15,15] \text { by }[-15,15]$$

The graph of \(r=a \theta\) in polar coordinates is an example of the spiral of Archimedes. With your calculator set to radian mode, use the given value of a and interval of \(\theta\) to graph the spiral in the window specified. $$a=-1,0 \leq \theta \leq 12 \pi,[-40,40] \text { by }[-40,40]$$

A painter is going to apply a special coating to a triangular metal plate on a new building. Two sides measure \(16.1 \mathrm{m}\) and \(15.2 \mathrm{m}\). She knows that the angle between these sides is \(125^{\circ} .\) What is the area of the surface she plans to cover with the coating?

Consider the following complex numbers, and work in order. $$ w=-1+i \quad \text { and } \quad z=-1-i $$ Multiply \(w\) and \(z\) using their rectangular forms and the FOIL method. Leave the product in rectangular form.