Precalc final

2 planes are perpendicular if

(a,b,c) = k (a,b,c)

Quadrantal angle values

(cos ø, sin ø) 0 = (1, 0) 90 (pi/2) = (0, 1) 180 (pi) = (-1, 0) 270 (3pi/2) = (0, -1)

Different ways to solve triangles (?)

(given 2 sides, angle between) 1. Use law of cosines for the missing side 2. Use law of sines to solve for the smallest remaining angle 3. Subtract from 180 degrees to find 3rd angle (given 3 sides) 1. Use law of cosines to solve for the largest angle 2. Use law of sines to solve for the smallest remaining angle

Polar coordinates

(r, ø) r = the distance (positive OR negative) (if negative just go diagonal in the other direction) ø = the angle (radians OR degrees)

Vector equation

(x, y) = (x0, y0) + t(a, b) x0 and y0: Any point we know is on the line a and b: directional vector

If t is a unit of time

(x, y) = (x0, y0) + t(a, b) x0 and y0: The location of the object at t=0 or intial loaction a and b: velocity vector t= time unit The magnitude (or speed) of the vector is square root of a^2 + b^2

Circle equation

(x-h)^2 + (y-k)^2 = r^2

Ellipse equation

(x-h)^2/ a^2 + (y-k)^2/ b^2 =1 A always biggest b^2 = a^2 - c^2

Hyperbola

(x-h)^2/ a^2 - (y-k)^2/ b^2 = 1 A always positive b^2 = c^2 - a^2 Asymptotes y = +- b/a x (horizontal a) OR y = +- a/b x Other graph xy = k root of 2k are foci, root k is verticies If k>0, positive, if k<0, negative

To convert degrees, min, seconds to decimal degrees

- divide minutes by 60 - divide seconds by 3600 Round to 4 decimal places

Standard position angle

- its vertex is at the origin of a coordinate plane - its initial side lies along the positive x axis

To convert decimal degrees to degrees, min, seconds

- multiply decimal portion by 60 to convert to minutes - multiply decimal portion of minutes by 60 to convert to seconds - round to the nearest second

Longitude

0-180 degrees (from prime meridian) (vertical)

Latitude

0-90 degrees (from equator) (horizontal)

30, 60, 90 triangle

1 across from 30 Root 3 across from 60 2 across from 90

45, 45, 90 triangle

1 across from 45 1 across from 45 Root 2 across from 90

Degrees

1 min = 1' = 1/60 degree 1 second = 1" = 1/60 ' = 1/3600 degree

Tips for solving trig equations

1. If the equation is written in terms of ø and 2ø, use double angle identities to rewrite in terms of ø, then solve for ø 2. If the equation is written all in terms of 2ø, solve for 2ø, divide the answers by 2 to find ø 3. NEVER divide by a trig function unless we can guarantee it is not 0

properties of cross product

1. u x v = -(v x u) 2. u parallel to v if and only if u x v = 0 3. u x v is a nwe vector that is perpendicular to u and v 4. | u x v | (magnitude) = |u| |v| sin ø where ø is the angle between u and v (this is the area of the parallelogram. to find the triangle, x 1/2)

Properties of dot product

1. v dot u = u dot v 2. u dot u = |u|^2 3. k (u dot v) = (ku) dot v = u dot (kv) 4. u dot (v + w) = uv + uw

hyperbola and eccentricity

2 foci = 2 directrix e = c/a directrix +- a/e or a^2/c units from center

ellipse and eccentricity

2 foci = 2 directrix e = pf/pd = c/a for f= (0,c) the directrix will be at y = a/e = a^/c (-c makes it negative) it will be x = the same if it is on different axis will be +- that many units from center

Vector

A mathematical quantity that has two parts, magnitude and direction

Segment of a circle

A segment of a circle is a region that is bounded by a chord and the arc connecting the end points of the chord Area of segment: area of sector - area of the triangle

Oblique triangle

A triangle with no right angles

Which quadrants sin, cos, and tan are positive in

ASTC All students take calculus

Quadrantal angle

An angle whos terminal side lies on an axis (0, 90, 180, 270)

Rotation clockwise

Angle is negative

Rotation counter clockwise

Angle is positive

Direction

Can be given as a bearing (i think), as an angle with the horizontal center, or relative to the second vector

Parametric equations

Describe the effect on each variable ex (x, y) = (3, 1) + t(1, 3) x = 3 + t y = 1 + 3t Can be used to find the time and places where a vector equation crosses a line or curve (plug them into the x and ys of the equation)

Magnitude

Describes the length of the vector and can represent things such as speed, force, distance

Operations on vectors

Given vector v (x1, y1) and vector u (x2, y2) Addition: v + u = (x1 + x2, y1 + y2) Subtraction: v - u = (x1 - x2, y1 - y2) Scalar multiplication: kV = k (x1, y1) = (kx1, ky1)

Complex rectangular to polar

Given z = a + bi r = square root a^2 + b^2 tan^-1 b/a = ø z = r(cisø)

y = cscx and y = secx

Graph corresponding sin/cos as dotted curve Draw asymptotes where the sin/cos cross horizontal center Draw curves going away from max and min

Complex number plane

Horizontal axis is the real number axis (a axis) The vertical axis is the imaginary axis (b axis)

Argand diagram and complex numbers

How to display complex numbers given z = a + bi, graph the point (a,b) on a complex number plane In an argand diagram, the length of the arrow drawn is called the absolute value of the complex number z, and it is denoted as |z| |z| = r = square root of a^2 + b^2 |z| is essentially r except |z| must always be positive

Do i need stuff about subtending

Idk

Bearings

Method 1: Given an angle measure from 0 to 360. With this method, the angles measured from due north rotating clockwise Method 2: Given a North or South directional, followed by an acute angle measure, followed by an east or west directional (N23 degrees W) remember: never eat soggy waffles

Solving equations with less than signs

Once solved, test between solutions in the regular equation. ex. ø = pi/12, 7pi/12, 13pi/12, 19pi/12 answer: 0<x<pi/12, 7pi/12<x<13pi/12, 19pi/12<x<2pi Test inbetween and it will be every other

Perimeter of a sector formula

P = 2r + s P is the perimeter r is the radius s is the arc

Periodic function: finding period, amplitude

P = smallest positive value of p A = max - min/2

y = sinx or y = cosx

Period: 2pi Amplitude: 1 (Sin starts at 0, cos starts at top) Cross horizontal center at: sinx: beginning, middle, end Cosx: 1/4 way, 3/4 way Reaches max at: Sinx: 1/4 way through Cosx: beginning and end

y = tanx or y = cotx

Period: pi Amplitude: n/a ***tangent is only function whose period is centered on y axis Will have asymptotes at beginning/end of period Will be one unit above horizontal center e: Tanx: 3/4 way through Cotx: 1/4 way through Cross horizontal center in middle

Reference angles in each quadrant

Q1 = the angle itself Q2 = 180 - ø Q3 = ø - 180 Q4 = 360 - ø

Complex numbers can be written in rectangular or polar form

Rectangular: z = a +bi Polar: z = r(cisø)

co-function identities

Sin A = cos (90 - A) Tan A = cot (90 - A) Sec A = csc (90 - A)

Trig reciprocal identities

Sin ø = 1/ csc ø Tan ø = 1/ cot ø Cos ø = 1/ sec ø

What sin, cos, tan are

Sin ø = opp/hyp or y/r Cos ø = adj/hyp or x/r Tan ø = opp/adj or y/x

Trig quotient identities

Tan ø = sin ø/ cos ø Cot ø = cos ø/ sin ø

(different version) Two vectors are parallel if

The lines containing them are parallel

Linear velocity formula

The rate at which the arc length changes v = s/t (also v = Ør/t, v = wr) v is linear velocity s is sector t is time

Angular velocity

The rate at which the central angle changes w = Ø/t w is angular velocity Ø is the central angle t is the time

Terminal side

The ray after rotation

Initial side

The ray prior to the rotation

Herons formula

The semi perimeter, s, is half the perimeter s = 1/2 (a+b+c) k = square root of ( s(s-a)(s-b)(s-c) )

Cramers rule

The solution of a system of equations is fractions (dx/d, dy/d, dx/d) d will be the regular 3x3 dx will be the 3x3 with the solutions in the irst column dy with solutions in second column dz with solutions in third column

Coterminal angles

Two angles are coterminal if they have the same terminal side

To find the component form of a vector from point A (x1, y1) to point B (x2, y2)

Vector AB = (x2 - x1, y2 - y1) The magnitude of the vector will be = square root of (x2 - x1)^2 + (y2 - y1)^2

The ambiguous case of the law of sines

When we are given an acute angle measure, and the side opposite the angle is less than the side adjacent, then one of these three things can occur 1. No triangle exists (sinb > 1) 2. We can get one right triangle (sinb = 1) 3. There are two triagnles (sinb < 1) (b is the only angle we have)

Scalar multiple of a vector

Will have the same direction if the scalar is positive, and the opposite direction if negative If we multiply by a scalar K, then the magnitude will be multiplied by |k|

Complex numbers

a +bi

All trig functions can be written in the form y = a func b (x-h) + k

a = changes amplitude (sinc/cos - amp = |a|)(tan, cot chnages the 1/4 points) b = changes the perid (sin/cos - period = 2pi/b) (tan/cot - period = pi/b) h = shifts horizontally k = shifts vertically

2x2 matrix

a1 a2 b1 b2 determinant just has straight bars around it determinant is a1b2-a2b1

The determinant of a 3x3 matrix

a1 a2 a3 b1 b2 b3 c1 c2 c3 1. solve by minors (take a row or column and solve by each ones minor. the minor will be the 2x2 matrix that doesnt include the row or column it is in) 2. Diagonals (rewrite the first two columns on the side. go diagonal to the right first multiply them then add those three, then subtract the other three diagonals) 3. row reduction (add to rows or columns (if u choose to do a row, add columns, viceversa) to get one 1 and the rest zeros then expand by minors) for 1. and 3. remember + - + - + - + - +

Law of cosines

a^2 = b^2 + c^2 - 2bc cosA or CosA = b^2 + c^2 - a^2/ 2bc

Planes equation

ax + by + cz = d (so long as a, b, and c are not all 0s) When given a, b, c find the intercepts to get the plane

The equation of a plane that is perpendicular to a non zero vector

ax + by + cz = d, where d = ax0 + by0 + cz0 ex Find equation of plane perpendicular to vector (5, 3, 1) at point (3, 2, -1) plug (5, 3, 1) into a,b,c and point into x,y,z solve for d theres the equation

Identifying equations from ax^2 + bxy + cy^2 +Dx + Ey + F = 0

b^2 - 4ac < 0, ellipse or circle (circle a =c, b= 0) b^2 - 4ac = 0, parabola b^2 - 4ac > 0, hyperbola

Half angle identities

cos x/2 = +- square root of 1 + cosx/2 sin x/2 = +- square root of 1 - cosx/2 tan x/2 = +- square root of 1 - cosx/ 1 + cosx tan x/2 = sinx / 1 + cosx tan x/2 = 1- cosx/ sinx

Finding the angle between vectors

cos ø = u dot v/ |u| * |v|

conic sections and eccentricity

e= PF/PD (p is a point on the plane) PF = the distance from p to the focus PD = The perpendicular distance from P to the directrix 0<e<1, conic is ellipse e=1, conic is parabola e>1, conic is hyperbola say given eccentricity 1/2 1/2 = pf/pd pd = 2pf plug into the equation with the distance to the directrix equal to distance to the foci from x,y to find equation

Polar to rectangular coordinates

given (r, ø) convert by (rcosø, rsinø)

Rectangular to polar coordinates

given (x, y) convert by r = square root x^2 + y^2 tan^-1 y/x = ø

Complex polar to rectangular

given z = r(cisø) z = r(cosø + sinøi) Solve cosø and sinø Multiply each by r z = a + bi

the cross product of vector U (a1, b1, c1) and vector V (a2, b2, c2)

i j k a1 b1 c1 a2 b2 c2 first vector always on top

Area of a triangle

k = 1/2bh k = 1/2absinc

Sector formula

k = 1/2sr s is arc r is radius k is sector area

Sectors

k = 1/2sr k= 1/2ør^2

The zero vector is

parallel and orthogonal to every vector (has magnitude of zero)

Orthogonal vectors

perpendicular vectors

Adding and subtracting vectors

place the initial point and the terminal point of the vectors together. The resultant is the initial point of the first vector, and the terminal point of the second to subtract just add the opposite vector

Trig sum and difference identities

recall cos (x+y) ≠ cosx + cosy cos (x+y) = cosxcosy - sinxsiny cos (x-y) = cosxcosy + sinxsiny sin (x+y) = sinxcosy + cosxsiny sin (x-y) = sinxcosy - cosxsiny tan (x+y) = tanx + tany/ 1 - tanxtany tan (x-y) = tanx - tany/ 1 + tanxtany

How is an angle formed

rotating a ray about its end point

Formula for arc length

s = Ør s is arc Ø is central angle r is radius

the angle between two planes

same as angle between two vectors

Vectors in polar component form

same as previous ones going to polar

Relationships with negative angles

sin (-ø) = -sinø cos (-ø) = cosø tan (-ø) = -tanø

angle between 3 dimensional vectors

sin ø = |u x v|/|u| |v|

Double angle identities

sin2x = 2sinxcosx tan2x = 2tanx/1 - tan^2x cos2x = cos^2x - sin^2x cos2x = 2cos^2x - 1 cos2x = 1 - 2sin^2x

Law of sines

sinA/a = sinB/b = sinC/c (keep unknown on top) or a/sinA = b/sinB = c/sinC

Pythagorean identities

sin^2ø + cos^2ø = 1 1 + tan^2ø = sec^2ø 1 + cot^2ø = csc2ø

Rewriting sums as products

sinx + siny = 2sin (x+y/2) cos (x-y/2) sinx - siny = 2cos (x+y/2) sin (x-y/2) cosx + cosy = 2cos (x+y/2) cos (x-y/2) cosx - cosy = -2cos (x+y/2) cos (x-y/2) i think whale said this isnt on it?

One to one functions

test with horizontal line test (y = sinx will be a one to one function)

the area of the parallelogram formed by vector v and vector u

the absolute value of the 2x2 determinant

the area of the parallelpiped defined by vectors vectors v, u and w

the absolute value of the 3x3 determinant

inclination of a line

the angle, with 0 ≤ angle < 180 that the line makes with the positive x axis m = tanx

Two vectors are perpendicular if

the lines that contain them are perpendicular

3 dimensional vectors are

the same as two dimensional

Resultant

the sum of two vectors

vector v is perpendicular to vector u if

their dot product equals zero

Two vectors are parallel if

they have the same direction, or opposite directions

Two vectors are equal vectors if

they have the same magnitude and direction

Two vectors are opposite vectors if

they have the same magnitude and opposite directions (vector a = - vector b)

Component form of a vector

v (with a line across): (x, y)

The dot product of vector V and vector U

v dot u = x1x2 + y1y2

Velocity describes

what happens every unit of time

Parabola

y - k = 1/4p (x-h)^2 (up, negative would be down) OR x - h = 1/4p (y-k)^2 (right, negative would be left) distance between directrix and focus is 2p

Shifts to y= f(x)

y = - f(x). Reflected over x-axis (horizontal center) y = c f(x). Stretched/shrinked vertically (ADD VALUES) (changes amp to CA) y = f(cx). Stretched/shrinked horizontally (ADD VALUES) (changes period to P/C) y = f (x-h) + k. Shift horizontally h units, vertically k units try to find the numbers that make it bigger/smaller

inverse trig functions

y = sin -1 (root2/2). Use that to find reference angle and use quadrant to find that angle sinx {x| -pi/2 ≤ x < pi/2} q1, q4 cosx {x| 0 ≤ x < pi} q1, q2 tanx {x| -pi/2 < x < pi/2} q1, q4 cotx {x| 0 < x < pi} q1, q2 secx {x| 0 ≤ x < pi/2, pi/2 < x ≤ pi} q1, q2 cscx {x| -pi/2 ≤ x < 0, 0 < x ≤ pi/2} q1, q4

Directional vector determines the slope of the line/vector

y/x (make sure to simplify/take out the scalar to get the slope)

x, y, z graph

z goes up x towards you y across

The product of two complex numbers

z1 x z2 = (r1 x r2) cis (ø1 + ø2)

Powers of complex numbers (Demoivres theorem)

z^n = r^n cis nø

Radians

Ø = s/r Ø is central angle S is arc R is radius (Radians have no units)