Pre-Calc Unit 𓆽𝟚

Convert to polar form: (x - 1)^2 + (y + 1)^2 = 2

(x - 1)^2 + (y + 1)^2 = 2 (x^2 - 2x + 1) + (y^2 + 2y + 1) = 2 x^2 + y^2 - 2x + 2y = 0 r^2 = 2x - 2y r^2 = 2r cos θ - 2r sin θ r = 2 cos θ - 2 sin θ

Sketching Graph of a Polar Equation Ex. r = 3 Ex. θ = π/3 Ex. r = 2 sin θ

- convert to rectangular form Ex. r = 3 - x^2 + y^2 = 9 (circle w/radius 3) Ex. θ = π/3 - straight line, passes through origin and makes angle π/3 w/polar axis - tan θ = y/x, tan π/3 = √3 - y/x = √3, y = x√3 Ex. r = 2 sin θ - use equation to determine polar coordinates of several points on curve - *(0, 0)* (π/6, 1) (π/4, √2) (π/3, √3) *(π/2, 2)* (2π/3, √3) (3π/4, √2) (5π/6, 1) *(π, 0)* - plot points on polar grid and connect

Lemniscates

- figure-eight-shaped curves - *r^2 = a^2 sin 2θ*: lemniscate that looks like diagonal "8" in 1st and 3rd Quads - *r^2 = a^2 cos 2θ*: lemniscate that looks like horizontal "8" through x-axis

Cardioid

- heart-shaped curve - any equation of form r = a(1 ± cos θ) or r = a(1 ± sin θ) - usually butt is on (0, 0)

plotting polar coordinates

- take given point (r, θ) and make length of segment r and draw at correct θ - +r means draw base arrow on right (pos. x-axis) - -r means draw base arrow on left (negl x-axis) - draw θ in relation to base arrow

Cartesian System

- x-y coordinate system - describes how we move horizontally and vertically using coordinates P(x, y)

Types of Polar Intersections

1. simultaneous intersection - polar graphs intersect at same time - solve for using algebra (most intersections) 2. Asynchronous intersection - polar graphs intersect at different times (one intersects spot first then the other) - Ex. sin circle STARTS at (0, 0) vs. cos circle ENDS at (0, 0) - Ex. (0, 0)

Hyperbola not centered at origin

foci on the x-axis: - graph opens left/right - equation: *(x-h)^2/a^2 - (y-k)^2/b^2 = 1* (a > 0, b > 0) - center: (h, k) - vertices: (h ± a, k) - co-vertices: (h, k ± b) - transverse axis: horiz., length 2a - asymptotes: (y - k) = ±(b/a)(x - h) (diagonal lines, solve for y to get equation of line) - foci: (h ± c, k), a^2 = c^2 - b^2 foci on the y-axis: - graph opens up/down - equation: (y-k)^2/a^2 - (x-h)^2/b^2 = 1 (a > 0, b > 0) - vertices: (h, k ± a) - co-vertices: (h ± b, k) - transverse axis: vert., length 2a - asymptotes: (y - k) = ±(a/b)(x - h) (diagonal lines, solve for y to get equation of line) - foci: (h, k ± c), a^2 = c^2 - b^2

convert to rectangular form: r = 2 sin (θ + π/4)

r = 2 sin (θ + π/4) r = 2 (sin θ cos π/4 + cos θ sin π/4) r = 2 (√2/2 sin θ + √2/2 cos θ) r = √2(sin θ + cos θ) r = √2(y/r + x/r) - x = r cos θ => x/r - y = r sin θ => y/r r^2 = √2x + √2y x^2 + y^2 = √2x + √2y (x^2 - √2x + (√2/2)^2) + (y^2 - √2y + (√2/2)^2) = (√2/2)^2 + (√2/2)^2 (x - √2/2)^2 + (y - √2/2)^2 = 1

convert to rectangular form: r = 4 csc(θ + π/6)

r = 4 csc(θ + π/6) r = 4 / sin(θ + π/6) r sin(θ + π/6) = 4 r (sin θ cos π/6 + cos θ sin π/6) = 4 - sin (x + y) = sin x cos y + cos x sin y r(sin θ • √3/2 + cos θ • 1/2) = 4 √3y + x = 8 y = 8√3/3 + √3x/3

Limaçons

- *r = a ± b sin θ* - *r = a ± b cos θ* - (a > 0, b > 0) - orientation depends on trig function (sin or cos) and sign of b - left/right orientation => r = a ± b cos θ - up/down orientation => r = a ± b sin θ - *a < b*: Limaçon w/inner loop => loop size is b-a (a + b is total length after loop) - *a = b*: cardioid => a is distance from intercept(s) to butt => 2b is amplitude (butt to single intercept) - *a > b*: dimpled limaçon - *a ≥ 2b*: convex limaçon

polar (r, θ) to rectangular (x, y) coordinates Ex. (5, 2π/3) to rectangular coordinates

- triangle w/x, y, r - sin θ = y/r => *y = r sin θ* - cos θ = x/r => *x = r cos θ* Ex. 1. (r, θ) = (5, 2π/3) - x = 5 cos 2π/3 = 5(-1/2) = -5/2 - y = 5 sin 2π/3 = 5(√3/2)= 5√3/2 2. (x, y) = (-5/2, 5√3/2)

Convert to polar form: x = y^2

x = y^2 r cos θ = r^2 sin^2 θ r = cos θ/sin^2 θ r = cot θ • csc θ

convert to rectangular form: a) r = 5 sec θ b) r = 2 sin θ c) r = 2 + 2 cos θ d) r = sin(θ - π/4)

- see what the graph of the equation going to be (line, parabola, circle, ellipse, hyperbola) to see what end form should look like when all variables are in x + y form a) r = 5 sec θ - line - r = 5/cos θ => x = 5 (x = r cos θ) b) r = 2 sin θ - circle - r^2 = 2r sin θ => x^2 + y^2 = 2y => x^2 + y^2 - 2y + 1 = 1 - r^2 = x^2 + y^2 - mult. • r, use above equation - bring variables onto one side to complete square => x^2 + (y - 1)^2 =1 c) r = 2 + 2 cos θ - cardioid - r^2 = 2r + 2r cos θ => x^2 + y^2 = 2r + 2x => x^2 - 2x + y^2 = 2r => (x^2 - 2x + y^2)^2 = 4r^2 => (x^2 - 2x + y^2)^2 = 4(x^2 + y^2) d) r = sin(θ - π/4) - ellipse - r = (sin θ cos π/4) - (cos θ sin π/4) (trig identity) => r = √2/2 sin - √2/2 cos θ => x^2 + y^2 = √2/2y - √2/2x => (x^2 + √2/2x + 1/8) + (y^2 - √2/2y + 1/8) = 1/4 => (x + √2/4)^2/(1/4) + (y - √2/4)^2/(1/4) = 1

convert each equation to polar form a) x^2 = 4y b) 4x - 7y = 2 c) x^2 + y^2 - 8y = 0

a) x^2 = 4y - parabola - (r cos θ)^2 = 4(r sin θ) => r^2 cos^2 θ = 4 r sin θ => r cos^2 θ = 4 sin θ => r = 4 sin θ / cos^2 θ => r = 4 tan θ • sec θ b) 4x - 7y = 2 - line - 4(r cos θ) - 7(r sin θ) = 2 => r(4 cos θ - 7 sin θ) = 2 => r = 2/4 cos θ - 7 sin θ c) x^2 + y^2 - 8y = 0 - ellipse - r^2 - 8y = 0 => r^2 = 8(r sin θ) => r = 8 sin θ

Ellipse not centered at origin

horizontal major axis (a > b > 0): - *(x-h)^2/a^2 + (y-k)^2/b^2 = 1* - center: (h, k) - vertices: (h ± a, k) - co-vertices: (h, k ± b) - foci: (h ± c, k) vertical major axis (b > a > 0): - *(x-h)^2/b^2 + (y-k)^2/a^2 = 1* - center: (h, k) - vertices: (h, k ± a) - co-vertices: (h ± b, k) - foci: (h, k ± c) b^2 + c^2 = a^2 **circle if you write equation in standard form and x^2 + y^2 have no denominators (a = b)

Common Polar Curves

- circles, spiral - Limaçons - roses - Lemniscates

Convert to rectangular form: r^2 = 2/sin 2θ

r^2 = 2/sin 2θ r^2 = 2/2 sin θ cos θ 1 = 1/r sin θ • r cos θ 1 = 1/xy y = 1/x

convert to rectangular form: tan θ = 2

tan θ = 2 (tan θ = y/x) => y/x = 2 y = 2x

Weird polar graphs

- plot points on a sin or cos graph - plot amplitude number firsts, then note vertical shift by comparing range => roses: lengths of ALL types of petals added up/2 => lemniscate-things: distance between x-ints/2 - note if period is still 2π or if its its shorter by looking at linear graph (e.g π, instead of writing r = a sin θ, write r = a sin 2θ) - if you have something that looks like an inner loop Limaçon flipped out, use r = a sin/cos θ but take absolute value

Find the polar equation from the graph

1. determine what type of polar equation - circles, spiral - Limaçons - roses - Lemniscates 2. take into account any shifts/stretches/reflections - Limaçons (a = b or e.g cardioid): total length from top of circle to butt of circle/2 = |a| - Limaçons (a < b e.g inner loop): total length from top of circle to butt of circle + loop length/2 = |a| - shifts = distance from x or y-intercept to butt - Limaçons r = a - b sin/cos θ if graph orientated left or down - Limaçons (a > b e.g dimpled limaçon): to make sure a > b, take |a| and subtract and shift down from (0, 0) of dimple (see wrk sht)

graph of y = a sin b(x - h) + k/ y = a cos b(x - h) + k

- *amplitude: |a|* ~ how high/low graph goes - *period: 2π/b* - b ~ 2π/period - apply same transformations (vert. stretch/compress, hoz. stretch/compress, shifts) to parent function sin x and cos x values - factor out "b" (if not already there) to find the period and hoz. shift (completely factor whats in parenthesis) - Period = 2π/b - sin t values (x, y): (0º, 0), (90º, 1), (180º, 0), (270º, -1), (360º, 0) OR convert degrees to radians - cos t values (x, y): (0º, 1), (90º, 0), (180º, -1), (270º, 0), (360º, 1) etc. OR convert º's to radians - if a vert. shift up/down, move "new" x-axis" up as well (instead of 0º, 0 etc., y-value will be replaced w/new y-value of shifted x-axis)

Roses

- *r = a sin nθ* => engulf axis - *r = a cos nθ* => between axis - as a increases, petal size increases (larger amplitude) - as n increases, more petals (shorter period, larger frequency) - n-leaved if n is odd - 2n-leaved if n is even - a is amplitude/size - points at tips of petals at angles as intervals of 360º/n - *r = a cos 2θ*: 4-leaved rose - *r = a cos 3θ*: 3-leaved rose - between 2nd Quad, between 3rd Quad, engulfing right x-axis - *r = a cos 4θ*: 8-leaved rose - *r = a cos 5θ*: 5-leaved rose - if unsure how to graph, graph sin or cos graph first, plot peak and trough points on polar grid (connect points in order)

Circles and Spiral

- *r = a*: circle w/center (0,0) and radius "a" - *r = a sin θ*: circle where "a" is diameter and bottom most edge is at polar axis/pole (above/below x-axis) - e.g r = 2a sin θ: radius a, centered at (a, π/2) - above if a is pos, below if neg - *r = a cos θ*: circle where "a" is diameter and left most edge at polar axis/pole (left/right of y-axis) - e.g r = 2a cos θ: radius a, centered at (a, 0) - right if a is pos, left if neg - *r = aθ*: spiral - r = radius of each spiral - θ = angle of rotation as curve spirals

Polar System

- distance straight from origin to a point - e.g as the crow flys vs. up/down/left/right - uses distances to specify location of point in the plane - determines angle this segment makes w/pos. x-axis - P(r, θ) or P(r0, θ0) - set up system: fixed point O (*pole/origin*) and draw ray from O (*polar axis*) - each point assigned polar coordinates P(r, θ) where: - r: distance from O to P - θ: angle between polar axis and segment OP - if θ pos., measure ccw from polar axis (ccw rotation) - if θ neg., measure cw (cw rotation) - if r neg., P(r, θ) is r units from pole in opp. direction of θ

Trig Identities

- identity: equation that is true for all values - write same expression different ways *Reciprocal Identities* - csc x = 1 / sin x - sec x = 1 / cos x - tan x = sin x / cos x - cot x = 1 / tan x = cos x / sin x *Pythagorean Identities* - sin^2 x + cos^2 = 1 - tan^2 x + 1 = sec^2 - 1 + cot^2 x = csc^2 x *Even-Odd Identities* - sin(-x) = -sin x - cos(-x) = cos x - tan(-x) = -tan x *Cofunction Identities* - sin(π/2 - x) = cos x - cos(π/2 - x) = sin x - tan(π/2 - x) = cot x - cot(π/2 - x) = tan x - sec(π/2 - x) = csc x - csc(π/2 - x) = sec x *Addition and Subtraction Formulas* sin (s + t) = sin s cos t + cos s sin t sin (s - t) = sin s cos t - cos s sin t cos (s + t) = cos s cos t - sin s sin t cos (s - t) = cos s cos t + sin s sin t tan (s + t) = tan s + tan t / 1 - tan s tan t tan (s - t) = tan s - tan t / 1 + tan s tan t *Double-Angle Formulas* sin 2x = 2 sin x cos x cos 2x = cos^2x - sin^2x cos 2x = 1 - 2 sin^2x cos 2x = 2 cos^2x - 1 tan 2x = 2 tan x / 1 - tan^2x *Reduction Identities* sin (x + π) = -sin x sin (x + π/2) = cos x cos (x + π) = -cos x cos (x + π/2) = -sin x tan (x + π) = tan x tan (x + π/2) = -cot x *Half-Angle Formulas* sin A/2 = ±√1 - cos A/2 cos A/2 = ±√1 + cos A/2 tan A/2 = 1 - cos A / sin A = sin A/ 1 + cos A tan A/2 = ±√1 - cos A / 1 + cos A

rectangular coordinates (x, y) to polar coordinates (r, θ) Ex. convert (-1, 1) to polar coordinates

- pythag: P(x, y) where x^2 (horz. distance) + y^2 (height) = r^2 (radius) - triangle w/x, y, r - *x^2 + y^2 = r^2* - *tan^-1 (y/x) = θ* - or *tan θ = y/x* - note what quadrant (x, y) is in to help find θ Ex. (x, y) = (-1, 1) 1. 1 + 1 = r^2 => r = √2 2. tan^-1(1/1) = θ => θ = 3π/4* 3. (r, θ) = (√2, 3π/4)

find polar coordinates for point Q w/r > or < 0

- r > 0: just convert (x, y) to regular polar coordinates (x^2 + y^2 = r^2, tan^-1 (y/x) = θ) - r < 0: the point directly across from r > 0 on unit circle - polar coordinates w/-r and find same θ but instead of using pos. x (polar) axis, use neg. (left) x-axis - e.g distance from 0 or 2π to original π is same as distance from π to new π - e.g π + old θ but make sure new point is in right quadrant - if original polar coordinate in Quad 1, r < 0 in Quad 3 (Quad 2 + Quad 4, Quad 3 + Quad 1, Quad 4 + Quad 2)

Weird polar graphs: roses Ex. rose w/12 alternating 1 unit and 5 unit petals

- recall rose form: - *r = a sin nθ* => engulf axis - *r = a cos nθ* => between axis graph to equation: 1. decide if it looks sin or cos (graph first point on sin/cos graph axis) 2. amplitude (a) = lengths of ALL types of petals added up/2 3. n = # petals TOTAL/2 4. find shifts => - if given a sin/cos graph, 1 petal is segment of graph above, one below, one above etc. x-axis Ex. linearly graph rose: first plot amplitude points (e.g 3, -3 or 6/2), then plot petal heights (1 and 5) - how far up did shift to reach 5? 2 units (+2 vertical shift) **when in doubt plot polar points on sin graph w/0, π/2, π, 3π/2, 2π

Parabola not centered at origin

- shift x^2 = 4py to have vertex (h, k): *(x - h)^2 = 4p(y - k)* *(y - h)^2 = 4p(x - k)* - find vertex first (h, k), then p, then focus, then directrix - add zeros if no h or k term (e.g (y + 0)^2 if given y) *vert. axis of symmetry (hoz. directrix) (x^2)* - parabola opens up if p > 0, downward p < 0 - focus: - x-value = x-value of vertex - set difference in y-values of vertex and focus point = p, solve for y-value of focus point - directrix: -p + y-value of vertex *horz. axis of symmetry (vert. directrix) (y^2)* - parabola opens right if p > 0, left p < 0 - focus point: - y-value = y-value of vertex - set difference in x-values of vertex and focus point = p, solve for y-value of focus point - directrix: -p + x-value of vertex

Polar Intersections

1. substitute - either solve for r then substitute or just substitute - recall trig identities - if ever have cos^2 θ or sin^2 θ, treat them like x^2 and use quadratic formula to factor 2. solve for sin θ or cos θ (depending on problem) - Ex. sin θ = -1 - use sin inverse to solve for θ (Ex. θ = 3π/2) - solve for ALL θ's (make sure accounting for every quadrant) 3. write in (r, θ) form - solve for r: take values of sin θ or cos θ and plug them into original equations