8.4 Polar Coordinates: Graphs
The equation fails the symmetry test with respect to the lineand with respect to the pole. It passes the polar axis symmetry test.
Tests will reveal symmetry about the polar axis. The zero isand the maximum value is
The graph is a rose curve,even
8.7 Parametric Equations: Graphs
The graph of the parametric equations is in red and the graph of the rectangular equation is drawn in blue dots on top of the parametric equations.
8.1 Section Exercises
The altitude extends from any vertex to the opposite side or to the line containing the opposite side at a 90° angle.
When the known values are the side opposite the missing angle and another side and its opposite angle.
A triangle with two given sides and a non-included angle.
no triangle possible
The distance from the satellite to stationis approximately 1716 miles. The satellite is approximately 1706 miles above the ground.
445,624 square miles
8.2 Section Exercises
two sides and the angle opposite the missing side.
is the semi-perimeter, which is half the perimeter of the triangle.
The Law of Cosines must be used for any oblique (non-right) triangle.
8.3 Section Exercises
For polar coordinates, the point in the plane depends on the angle from the positive x-axis and distance from the origin, while in Cartesian coordinates, the point represents the horizontal and vertical distances from the origin. For each point in the coordinate plane, there is one representation, but for each point in the polar plane, there are infinite representations.
Determinefor the point, then moveunits from the pole to plot the point. Ifis negative, moveunits from the pole in the opposite direction but along the same angle. The point is a distance ofaway from the origin at an angle offrom the polar axis.
The pointhas a positive angle but a negative radius and is plotted by moving to an angle ofand then moving 3 units in the negative direction. This places the point 3 units down the negative y-axis. The pointhas a negative angle and a positive radius and is plotted by first moving to an angle ofand then moving 3 units down, which is the positive direction for a negative angle. The point is also 3 units down the negative y-axis.
A vertical line withunits left of the y-axis.
A horizontal line withunits below the x-axis.
8.4 Section Exercises
Symmetry with respect to the polar axis is similar to symmetry about the-axis, symmetry with respect to the pole is similar to symmetry about the origin, and symmetric with respect to the lineis similar to symmetry about the-axis.
Test for symmetry; find zeros, intercepts, and maxima; make a table of values. Decide the general type of graph, cardioid, limaçon, lemniscate, etc., then plot points at and sketch the graph.
The shape of the polar graph is determined by whether or not it includes a sine, a cosine, and constants in the equation.
symmetric with respect to the polar axis
symmetric with respect to the polar axis, symmetric with respect to the line symmetric with respect to the pole
symmetric with respect to the pole
inner loop/two-loop limaçon
inner loop/two-loop limaçon
inner loop/two-loop limaçon
They are both spirals, but not quite the same.
Both graphs are curves with 2 loops. The equation with a coefficient of has two loops on the left, the equation with a coefficient of 2 has two loops side by side. Graph these from 0 toto get a better picture.
When the width of the domain is increased, more petals of the flower are visible.
The graphs are three-petal, rose curves. The larger the coefficient, the greater the curve’s distance from the pole.
The graphs are spirals. The smaller the coefficient, the tighter the spiral.
and at sinceis squared
8.5 Section Exercises
a is the real part, b is the imaginary part, and
Polar form converts the real and imaginary part of the complex number in polar form using and
It is used to simplify polar form when a number has been raised to a power.
8.6 Section Exercises
A pair of functions that is dependent on an external factor. The two functions are written in terms of the same parameter. For example,and
Choose one equation to solve forsubstitute into the other equation and simplify.
Some equations cannot be written as functions, like a circle. However, when written as two parametric equations, separately the equations are functions.
answers may vary:
answers may vary: ,
8.7 Section Exercises
plotting points with the orientation arrow and a graphing calculator
The arrows show the orientation, the direction of motion according to increasing values of
The parametric equations show the different vertical and horizontal motions over time.
There will be 100 back-and-forth motions.
Take the opposite of theequation.
The parabola opens up.
approximately 3.2 seconds
8.8 Section Exercises
lowercase, bold letter, usually
They are unit vectors. They are used to represent the horizontal and vertical components of a vector. They each have a magnitude of 1.
The first number always represents the coefficient of theand the second represents the
a. 58.7; b. 12.5
4.635 miles, 17.764° N of E
17 miles. 10.318 miles
Distance: 2.868. Direction: 86.474° North of West, or 3.526° West of North
4.924°. 659 km/hr
21.801°, relative to the car’s forward direction
parallel: 16.28, perpendicular: 47.28 pounds
19.35 pounds, 231.54° from the horizontal
5.1583 pounds, 75.8° from the horizontal
distance of the plane from point2.2 km, elevation of the plane: 1.6 km
symmetric with respect to the line
- The ball is 14 feet high and 184 feet from where it was launched.
- 3.3 seconds