Chapter 9: Polar Coordinates, Vectors

Show that the graph of the equation $r\cos \theta =a$is a vertical

line a units to the right of the pole if $a\ge 0$and $\left|a\right|$units to

the left of the pole if $a<0$.

The letters r and $\theta$represent polar coordinates. Write each equation using rectangular coordinates $\left(x,y\right)$.

role="math" localid="1646747340400" $r=\frac{3}{3-\cos \theta }$

Weight of a Car A magnitude of 1200 pounds of force is required to prevent a car from rolling down a hill whose incline is $15°c$to the horizontal. What is the weight of the car?

Correct Direction for Crossing a River A river has a constant current of $3km/hr$. At what angle to a boat dock should a motorboat capable of maintaining a constant speed of$20km/hr$ be headed in order to reach a point directly opposite the dock? If the river is $\frac{1}{2}$ kilometer wide, how long will it take to cross?

Show that the graph of the equation $r=2a\sin \theta$, $a>0$,

is a circle of radius a with center at $(0,a)$in rectangular

coordinates.

Chicago In Chicago, the road system is set up like a Cartesian plane, where streets are indicated by the number of blocks they are from Madison Street and State Street. For example, Wrigley Field in Chicago is located at $1060$West Addison, which is $10$blocks west of State Street and $36$ blocks north of Madison Street. Treat the intersection of Madison Street and State Street as the origin of a coordinate system, with east being the positive x-axis.

(a) Write the location of Wrigley Field using rectangular coordinates.

(b) Write the location of Wrigley Field using polar coordinates. Use the east direction for the polar axis. Express $\theta$in degrees.

(c) U.S. Cellular Field, home of the White Sox, is located at $35$th and Princeton, which is $3$blocks west of State Street and $35$blocks south of Madison. Write the location of U.S. Cellular Field using rectangular coordinates.

(d) Write the location of U.S. Cellular Field using polar coordinates. Use the east direction for the polar axis. Express $\theta$in degrees.

Finding the Correct Compass Heading The pilot of an aircraft wishes to head directly east but is faced with a wind speed of $40mi/hr$from the northwest. If the pilot maintains an airspeed of $250mi/hr$, what compass heading should be maintained to head directly east? What is the actual speed of the aircraft?

Show that the graph of the equation $r=-2a\sin \theta$, ,

is a circle of radius a with center at $(0,-a)$in rectangular

coordinates.

Show that the formula for the distance d between two points $P_1=(r_1,{\theta }_1)$and $P_2=(r_2,{\theta }_2)$ is role="math" localid="1646753790944" $d=\sqrt{{r_1}^2+{r_2}^2-2r_1{r}_2\cos ({\theta }_2-{\theta }_1)}$.

Show that the graph of the equation $r=2a\cos \theta$, $a>0$

is a circle of radius a with center at $(a,0)$in rectangular

coordinates.