Pre-cal
angle
(amt of rotation) Rotation of a ray about its endpoint
problem solving: circles What is the equation of the circle with center at (4, -5) and a point on the circle at (4, -2)? What is the radius?
(x−4)²+(y+5)² =9 r=3
Co-terminal angles
-angles with the difference of 360 -angles with the same terminal side
convert from radian to degree: 4pi/7
102.857°
convert from degree to min and sec: 104.36°
104° 21' 36"
convert from degree to min and sec: 14.333°
14° 19' 58.8"
radian to degree
180°/pi
The ends of a rope are held in place at the top of two posts, 9m apart and each one 8m high. If the rope assumes a parabolic shape and touches the ground midway between two posts, how high is the rope 2m from one of the posts
2.47m
convert from min and sec to degree: 24° 12' 6"
24.24°
convert from degree to radian: 312°
26pi/15
convert from radian to degree 7pi/3
420°
convert from degree to radian: 215°
43pi/36
convert from min and sec to degree: 74° 8' 12"
74.14°
Quadrantal angles
90°, 180°, 270°, 360°
Problem solving: circles Find the center and radius of this circle: x²+y²−8x+2y−19=0
C (4, -1) r= 6
degree to radian
degree= pi/180°
Unit of measure for angles
degrees radian minutes seconds arch length
word problem: ellipse the lengths of the major and minor axes of an ellipse are 10cm and 8cm, respectively. find the distance between the foci and give the equation of the given ellipse
distance between foci= 6 equation of the ellipse: x²/25 + y²/16 = 1
Circle
generated when a plane intersects a cone perpendicular to the vertical axis or parallel to the base
Ellipse
generated when a plane intersects the cone not parallel to the base
Ellipse
locus of points where the sum of the distances from the two fixed points is a constant
Special angles
multiples of 30° and 45°
word problem: hyperbola assume that the hyperbola in the tower is modeled x²/36 - y²/225 = 1 with 160m as its height. Find the width at the top and its narrowest part in the middle. Also give its standard equation
narrowest part= 2a = 12m top width= 65.12m standard equation of the hyperbola: x²/36 - y²/225 = 1
Identify the conic section represented by the equation x²= 23 (y-2)
parabola
Hyperbola
parts of hyperbola: -Center -Focus/foci- c units away from the center (longest) -Vertices (transverse axis)- a units away from the center -Conjugate axis- 2b & perpendicular to the transverse -Auxiliary Rectangle -Asymptote -Hyperbola
3 undefined terms
point, line, plane
word problem: hyperbola an explosion is recorded by two microphones that are 1500 meters apart. microphone 1 received that sound 3 seconds before microphone 2. assuming sound travels at 300m/s. determine the possible locations of the explosion relative to the location of the microphone. what is the equation of the hyperbola being modeled by the situation
possible location of the explosion: anywhere near M1 or microphone 1 standard equation: x²/102 500 - y²/360 000 = 1
getting Arc length
s=rθ
Ellipse
standard equation (0,0) Horizontal: x²/a² + y²/b² = 1 Vertical: x²/b² + y²/a² = 1 standard equation (h,k) Horizontal: (x-h)²/a² + (y-k)²/b² Vertical: (x-h)²/b² + (y-k)²/a²
Circle
standard equation at any point in plane= (h-k)² + (y-k)²
Parabola
Standard Equation (0,0) x²= 4cy upward x²= -4cy Downward y²= 4cx to the right y²= -4cx to the left
Hyperbola
Standard equation (0,0) -Opening to the right and left: x²/a² - y²/b² = 1 -Upward and Downward: y²/a² - x²/b² = 1 Standard Equation (h,k) -Opening to the right and left: (x-h)²/a² - (y-k)²/b² = 1 -Upward and downward: (y-k)²/a² - (x-h)²/b² = 1
Parabola
Standard equation (h,k) (x-h)² = 4c (y-k)² upward (x-h)² = -4c (y-k)² downward (y-k)² = 4c (x-h)² to the right (y-k)² = -4c (x-h)² to the left
Circle
Standard equation at the origin= x² + y² = r²
Trigonometry
Study of Triangles
identify the conic section represented by the equation and its orientation 2x² + 5y² - 8x - 10y - 7 = 0
Ellipse: horizontal
Hyperbola
Formed when a plane intersects a cone perpendicular to the base. The plane is angled greater than the slant side
Ellipse
Forms a bounded curve
Parabola
Forms an unbounded Curve
Circle
General Form = Ax² + By² + Cx + Dy + E = 0 A=B
Ellipse
General form: Ax² + By² + Cx + Dy + E= 0 A and B have the same sign but are unequal
Parabola
Generated when a plane intersects one of the cone and is parallel to the slant side.
Hyperbola
Locus of points where the difference of the absolute distance from the two fixed point is a constant
Parabola
Locus of points wherein its distance from the focus is equal to the perpendicular distance to the directrix
Ellipse
Parts of Ellipse: Vertices (longest) Foci Co-Vertices Distance from center to Vertex = a Distance from center to focus= c Distance from center to Co-vertex= b Pythagorean theorem in ellipse: a² = b² + c²
Circle
Set of points equidistant from the fixed point called center
