In-Laws / Outlaws
When can we use Law of Sines to solve a problem and when do we have to use Law of Cosines?
Law of Sines can be used when we have a distance across from a given angle. Otherwise, we have to use Law of Cosines.
What is the definition of magnitude on a vector and how do we find it?
Distance from initial point to terminal point. We plot both points and use the Pythagorean Theorem (distance formula) to determine the distance.
What is the distance from the vertex to the focus on a parabola and how do we use it?
Known as "p." We use it (multiplied by 4) as the width of the parabola.
How do we find c for a hyperbola and for an ellipse?
Hyperbola: c2 = a2 + b2
Ellipse: c2 = a2 - b2
Determine two additional polar coordinates for the given point:
(-5, pi)
Two of:
(-5, 3pi) (5, 0) (5, 2pi)
Give the equations for both:
a. Law of Sines
b. Law of Cosinesa. a / sin A = b / sin B
b. c2 = a2 + b2 - 2ab cos C
Find the component form of the vector v.
Initial point: (-3, 7) Terminal point: (11, 16)
(14, -23)
If the focus is (-2, 3) and the directrix is x = -4, in what direction does the parabola open?
The parabola opens to the right.
Determine the conic that we can graph from each of the following:
a. x2 - 3x + 2y2 + 4y - 12 = 0
b. 3x2 + 4x - 5y2 + y = 24
c. x + 3y2 - 1 = 0
a. Ellipse
b. Hyperbola
c. Parabola
Change the following point from rectangular to polar coordinates:
(-6, -8)
(10, 53.13 degrees)
Find the area of the triangle:
A = 33 degrees
b = 7
c = 10
Area = 19.06 units2
Given u = (2, 7) and v = (-6, 5), find the resultant vector and determine its magnitude.
2u - 3v
vector: (22, -1)
Magnitude = 22.02
Determine the equation for the parabola:
Focus (4, -1), directrix: y = 3
Point on the parabola: (2, 0)
-4(y-1)=(x-4)2
Find the center, vertices, foci, and eccentricity of the ellipse.
9x2+ 4y2 + 36x - 24y + 36 = 0
Center: (-2, 3)
Vertices: (-2, 6), (-2, 0)
Foci: (-2, 3 +/- sqrt 5)
Eccentricity: sqrt 5 / 3
Convert the following points from polar to rectangular coordinates:
a. (2, 3pi/4)
b. (-4, -pi/3)
a. (-sq rt 2, sq rt 2)
b. (-2, 2sqrt3)
Solve the triangle:
a = 6, b = 7.3, c = 12.4
A = 19.12 degrees
B = 23.49 degrees
C = 137.39 degrees
Explain what determines whether two vectors are equivalent.
They must have both the same magnitude and the same direction.
Determine the equation for the parabola and give the focus:
vertex: (-3, 5), directrix: y = 8,
focus: (-3, 2)
-12(y-5)=(x+3)2
Find the equation, center, vertices, foci, and eccentricity of the hyperbola:
9x2 - 4y2 + 36y - 6x - 53 = 0
Center: (1/3, 9/2)
Vertices: (2 1/3, 9/2), (-1 2/3, 9/2)
Foci: (1/3 +/- sqrt 13, 9/2)
eccentricity: e = sqrt 13 /2
Convert the rectangular equation to polar form.
x = ar = a sec (theta)
An airplane flies 370 miles from point A to point B with a bearing of 24 degrees. Then it flies 240 miles from point B to point C on a bearing of 37 degrees. Find the distance and bearing from point A to point C.
606.3 miles;
Bearing: 29.1 degrees
Using the vectors u = (3, -3) and v = (-1, -5),
a. Sketch the resultant vector on a grid board: 2u-2v
b. Determine the magnitude of the resultant vector.
a. (Sketch) = (8, 4)
b. ll 2u-2v ll = 4 sqrt 5
There is a suspension bridge that is 1275 feet from tower to tower. The towers holding the bridge cable structure are 325 feet tall, and the cables come within 5 feet of the roadway at their lowest point.
What is the parabolic model for this bridge?
1270(y-5) = x2
Determine if the following is an equation of a parabola, a circle, an ellipse, or a hyperbola. Give reasons for your response.
100x2 + 100y2 -100x + 400y + 409 = 0
Circle
r = 2 sin (theta)
x2 + (y - 1)2 = 1
Circle with center of (0, 1), radius of 1