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