chapter 6

Question 1

law of sines works when given

Answer 1

asa
aas
ssa

Question 2

law of sines does not work when given

Answer 2

sas
ass

Question 3

ambiguous case possibilities

Answer 3

A is acute or obtuse

Question 4

A is acute in ass

Answer 4

solve for B

Question 5

A is acute in ass possinilities

Answer 5

B is undefined because b is shorter than heigt (0)
B is smaller than A (1)
B is larger than A (2) use B and 180-B

Question 6

A is obtuse in ass possibilities

Answer 6

a>b 1 triangle
a<b 0 triangles

Question 7

obtuse pythagorean

Answer 7

a^2+b^2<c^2

Question 8

acute pythagoran

Answer 8

a^2+b^2>c^2

Question 9

law of cosine can be used when

Answer 9

SSS
sas

Question 10

law of cosine sss

Answer 10

find largest angle first
js do this always tbh

Question 11

why in acute and obtuse c^2>/<a^2+b^2

Answer 11

equals means right triangle
and law of cos with -2abcosC means -cos
-cos is actually positive
cos in quadrant 2 is negative which is obtuse angle

Question 12

bearing

Answer 12

an acute angle formed by the north/south line

Question 13

expression for inside height of triangle

Answer 13

csinA just multiplu using law of sines and sin90

Question 14

remember A means

Answer 14

area

Question 15

vector

Answer 15

quantity with magnitude and direction

Question 16

component form

Answer 16

<x, y>

Question 17

linear combination form

Answer 17

xi + yj

Question 18

adjust for quadrant

Answer 18

Q2 180-
Q3 180+
Q4 360-

Question 19

finding component form dont add

Answer 19

just count

Question 20

converting parametric to rectangular

Answer 20

solve for t in one equation
substituteq

Question 21

make sure in parametrer

Answer 21

to state domain

Question 22

steps to writing set of parametric equations

Answer 22

y=mx+b is equivalent to x=at+b and y=ct+d
1. initial point create at t=0 and solve for b and d
2. second point at t=1 and solve for a and c

Question 23

making equayions with vectors

Answer 23

draw triangles and use cos and sin

Question 24

Front: Write parametric equations for
𝑦=√𝑥+9

Answer 24

Let the parameter equal the square root.
𝑡=𝑥+9
Square both sides:
𝑡^2=𝑥+9
Solve for 𝑥
𝑥=𝑡^2−9
Parametric equations:
𝑥=𝑡^2−9
x=t
y=t

Question 25

Front: Strategy for converting equations with square roots into parametric equations.

Answer 25

Back: Let the parameter equal the square root.

Question 26

How do you eliminate the parameter if x=e^t and y=e^3t?

Answer 26

Back: Solve for 𝑡 using x. x=e^t Take natural log: lnx=t Substitute into 𝑦 𝑦=𝑒3^(ln𝑥) Simplify: y=𝑥^3

Question 27

Front: Key rule for parametric equations involving 𝑒^𝑡

Answer 27

Back: Take natural log to isolate lnx=t Then substitute t into the other equation. Example: 𝑦=𝑒^(3ln𝑥)=𝑥^3

Question 28

Front: How do you eliminate the parameter if 𝑥=𝑡^2 and 𝑦=𝑡^3

Answer 28

Solve for 𝑡 from x 𝑥=𝑡^2 𝑡=root𝑥 Substitute into 𝑦

Question 29

Front: How do you eliminate the parameter if 𝑥=2^𝑡 and y=8^t

Answer 29

Back: Take log base 2 of both sides of the first equation. log2=x substitute into y y=8^(log2x) y=(2^3)^(log2x) use exponent rules y=(2^log2x)^3=x^3 y=x^3

Question 30

If x=a^t and y=a^kt

Answer 30

then y=x^k example x=2^t and y=2^3t y=x^3

Question 31

using points do both

Answer 31

equations

Question 32

domain when writing parametric new set

Answer 32

t gewater or equal to0

Question 33

(r, theta) and (-r, theta)

Answer 33

pi radians apart

Question 34

positive r but opposite angle

Answer 34

polt normally

Question 35

negative angle

Answer 35

dont flip quadrant

Question 36

polar equation r=x

Answer 36

circle with radius x

Question 37

theta = 5pi/4 describe

Answer 37

line neither horizontal or vertical because its going on that angle duhhhh

Question 38

mot a constant function in polar coords

Answer 38

straight lines bc not a constant radius or constant angle

Question 39

independent and dependent of polar

Answer 39

theta and r

Question 40

circle cos sym and petal

Answer 40

x axis, petal always on x axis

Question 41

circle sin sym and petal

Answer 41

y axis petal on y axis when n is odd

Question 42

small circle equations

Answer 42

r=acostheta and r=asintheta

Question 43

r=acostheta

Answer 43

a>0 right of y axis a<0 left of y axis polar axis summetry

Question 44

r=asintheta

Answer 44

a>0 above polar axis a<0 below polar acis y axis symmetry

Question 45

roses equations

Answer 45

r=acos(ntheta) and r=asin(ntheta_

Question 46

r=acos(ntheta)

Answer 46

polar axis symmetry 1st petal at theta=0

Question 47

r=asin(ntheta)

Answer 47

pi/2 symmetry firs petal at pi/2NNNNN

Question 48

LOOP

Answer 48

a

Question 49

catoid

Answer 49

a=b

Question 50

denr

Answer 50

a>b

Question 51

limacon sin sym and int

Answer 51

y and x

Question 52

limacon cos sym and int

Answer 52

x and y

Question 53

limacon

Answer 53

r=a+bcostheta or r=a+bsintheta

Question 54

tan unit circle

Answer 54

0, root3/3, 1, root 3, undefined

Question 55

why dented limacon never pass thru pole

Answer 55

sinc a>b, min radius will always be greater than zero

Question 56

Front: How do you check if a limacon passes through the pole?

Answer 56

The curve passes through the pole when: r=0 example: r=a+bcosthea set equal to zero 0=a+bcostheta and solve costheta=-a/b if this value is between -1 and 1, it passes thru pole

Question 57

When does a limacon have an inner loop (and pass through the pole)?

Answer 57

inner loop (passes through pole) cardioid (touches pole once) dent (does not reach pole)

Question 58

How do you check if a polar equation passes through a point (r,θ)?

Answer 58

Substitute the given θ into the equation. If the resulting value of r equals the given r, the point lies on the graph. Example: point (3,π) Plug in θ=π and check if r=3.

Question 59

How do you find the center and radius of a circle from a polar equation like 𝑟 = 𝑎 sin 𝜃 r=acosθ?

Answer 59

Multiply both sides by 𝑟 Use identities: r^2=x^2+y^2 rsinθ=y rcosθ=x Convert to rectangular form. Then identify the center and radius.

Question 60

fidnign type of symmetry

Answer 60

check for both

Question 61

Front: What is the smallest value of theta such that 9sin(4theta) = 9?

Answer 61

Back: Divide both sides by 9. 9sin(4theta) = 9 sin(4theta) = 1 Sine equals 1 at: 4theta = pi/2 Solve for theta: theta = (pi/2) / 4 theta = pi/8 Smallest value: theta = pi/8

Question 62

Front: Why does sin(4theta) = 1 give the location of the first petal of r = 9sin(4theta)?

Answer 62

A petal reaches its tip when r is maximum. For r = 9sin(4theta): The maximum value of sin(4theta) is 1. So the maximum radius is: r = 9 This happens when: sin(4theta) = 1 The smallest angle that satisfies this is: theta = pi/8 This angle gives the direction of the first petal.

Question 63

Front: How do you find the direction of the first petal of r = a sin(n theta)?

Answer 63

Back: The first petal occurs when the sine function is maximum. sin(n theta) = 1 So: n theta = pi/2 theta = pi / (2n) Example: r = 9sin(4theta) theta = pi / (2*4) theta = pi/8 This gives the direction of the first petal.

Question 64

Front: When does the first petal occur for r = a cos(n theta)?

Answer 64

Back: A petal occurs when r is maximum. For cosine, the maximum value is 1. cos(n theta) = 1 This happens when: n theta = 0 Solve: theta = 0 So the first petal of r = a cos(n theta) occurs at: theta = 0

Question 65

Front: Quick rule for first petal location of rose curves.

Answer 65

Back: r = a cos(n theta) First petal occurs at: theta = 0 r = a sin(n theta) First petal occurs at: theta = pi / (2n) Example: r = 9sin(4theta) theta = pi / (2*4) = pi/8

Question 66

vector graphing alwasyc check

Answer 66

signs

Question 67

rewriting polar equation of theta= as rectangular

Answer 67

find tan and thats m value/slope with no intercept

Question 68

converting polar to cartesion

Answer 68

dont add or subtract js multiply or diice

Question 69

reqriting eqautoin to polar

Answer 69

when you have r^2 make it r(r-yayayay) then divide two options out

Question 70

what as domain at Front: How do you find the domain (theta values) for polar equations?

Answer 70

The domain is all theta values where r is defined. Trig functions like sin(theta) and cos(theta) are defined for all real numbers. So most polar equations using sine or cosine have domain: 0 <= theta <= 2pi This interval traces the entire graph once.

Question 71

Front: What is the domain of theta for r = 3cos(theta)?

Answer 71

cos(theta) is defined for all theta. So the domain is: 0 <= theta <= 2pi This interval traces the full graph.

Question 72

domain of 0 2pi with equations

Answer 72

petals when n is even limacons? circles that look like r=a

Question 73

r=a

Answer 73

centedered at pole

Question 74

describing graph of r=a+betc when b=0

Answer 74

(0,a) (-a,0) (0,-a) (a,) and stuff

Question 75

Front: How can you tell if a limacon is flipped upside down?

Answer 75

Limacons are of the form: r = a + b sin(theta) → vertical orientation r = a + b cos(theta) → horizontal orientation Vertical limacons (sin): r = a + b sin(theta) → petal or bulge points up r = a − b sin(theta) → petal or bulge points down Horizontal limacons (cos): r = a + b cos(theta) → petal or bulge points right r = a − b cos(theta) → petal or bulge points left