Precalc Final
domain of ln(x+2)
(-2, infinity)
dot product a*a does not = a squared
(x1*x2+y1*y2)
find <B A: (2,0) B: (3,2) C: (-1,1)
*CosB= BC*BA/ ||BC||*||BA|| B=arccos (6/sqrt85)
Projection of AB on AC A: (2,0) B: (3,2) C: (-1,1)
*Projac(AB)= AB*AC/||AC||^2 *(AC) <3/10, -1/10>
find exact value of cos(2arcsin(1/3))
*a=arcsin(1/3) *cos2a *7/9
cos(arcsin(3/5)+arccos(1/2) find value
*a=arcsin(3/5) *b=arccos(1/2) *use cos(a+b) equ
what equation do you use when finding angles with arcs
*cosA=AB*AC/(||AB||*||AC||)
2sin(2x+pi/4)=-1
*divide by 2 sin=-1/2 --> 7pi/6, 11pi/6 *set these values = to # in (...) 2x+pi/4=7pi/6+2npi 2x+pi/4=11pi/6+2npi Ans: x=19pi/24+npi; x=11pi/24+npi
cos(2x)=sinx
*double angle formula 1-2sin^2=sinx 2a^2+a-1=0
hot air balloon rises vertically. Angle of elevation changes from 45 to 60 deg. 500 ft away. How fast is balloon rising?
*draw 2 separate triangles 500sqrt3-500ft
whats <A if: A:(1,2) B: (2,3) C: (4,0)
*find distance between AB, AC *cosA=AB*AC/(||AB||*||AC||) *A=arccos(1/sqrt26)
log3(x)+log3(x-8)-2=0
*log3(x)+log3(x-8)=2 *log3(x^2-8x)=2 x=9
3^(2x+1)=4^x-2)
*put ln on each side *break down *bring exponents in front *multiply exponents if more than 1 term *get x to same side and solve
5^(2x+1)=4^(x+2)
*put powers to ln x=-ln5+2ln4/(2ln5-ln4)
log3(tanx)=1/2
*tanx=3^(1/2) *pi/3+2npi
cos2x+sinx=0
*use double angle formula *1-2sin^2x+sinx *-2a^2+a+1 x=7pi/6+2npi x=11pi/6+2npi x=pi/2+2npi
sin(lnx)=0
*use e values x=e^(0+2npi) x=e^(pi+2npi)
sinA=3/5 in QII, secB=-13/5 in QIII Whats Sin(A+B)? what new Q is it in?
*use formula *(3/5)(-5/13)+(-4/5)(-12/13)= 33/65 Quadrent: *need to repeat in cos *both pos, so QI
cosx=-1/2 [pi, 3pi] solve for x
*values=2pi/3 and 4pi/3 ans: 4pi/3, 8pi/3
exact value of tan(5pi/8)
*when abnormal #, multiply by 2 *use 1/2 angle formula -sqrt[(1+sqrt2/2)/(1-sqrt2/2)]
tan(arcsin(-2/3))
*when arc-trig value, draw triangle -2sqrt5/5
when is something parallel
*when you multiply vectors together and it =0 ex <1,-4>*<4,1>=0
wind from S30W, 20 mph. Airplane is flying east, 30mph Which direction is plane flying? speed?
*wind: <10, 10sqrt3> (20cos30, 20sin30) *plane: <30, 0> *velocity: add together; <10, 10sqrt3>+<30,0>= <40, 10sqrt3> *Direction: ||velocity||= (sqrt(40^2)+(10sqrt3)^2)= 10sqrt19
sin(t+pi)
-sintx
sin(-2pi/3)
-sqrt3/2
0*a= a*a= mA*B=
0 |a|^2 A*mB
sec(2x-pi/4)=-2
11pi/24+npi or 19/24pi+npi
value of e
2.72
ln(x^2)=(lnx)^2
2lnx=(lnx)^2 2a=a^2
sinx=3/5 in Q III whats cos3x
3(sqrt7)/8
100^log5=
5^2... 25
arcsin(sin 5pi/4)
5pi/4 is in QIII distance to x is pi/4 ans: -pi/4
Proj of a onto b (ProjbA) a=<2,3> b=<1,0>
<2,0>
p=2000 12% quarterly how much after 5yrs?
A=2000(1+(.12/4))^4*5
parallel to AB A: (2,0) B: (3,2)
AB: <1,2> direction: <1/sqrt5, 2/sqrt5> **(run/hypotanuse, rise/hypotanuse) ANS: <-1/sqrt5, -1/sqrt5>
logAB= Log(A/B)=
LogA+logB LogA-LogB
arccos(cos4)
a=arccos(cos4) cosa=cos4 0<a<pi cos4--> 4-pi pi-(4-pi)= *2pi-4*
reference angle: cot(-959)
add 360 until you get 59. It lands in QII so its -cot(59)
sin(cos^-1(sin^-1 (0))
arcsin0=0 arccos0= pi/2 sinpi/2= 1
tanx=-1/2 solve on [4pi, 6pi]
arctan1/2 sol: 4pi-arctan(1/2); arctan(1/2)-5pi
arcsin(2)= no solution arcs(2)= pi/6 *inverse means second possible value (ex if in neg, only other solution would be other neg quadrant)
can't be greater or less than 1
cosx=1/2 whats cos4x?
cos(2*2x)=2cos^2(2x)-1.. =-1/2
cos(3x+pi/4)= -sqrt2/2
cos=-sqrt2/2 3x+pi/4=3pi/4 3x+pi/4=5pi/4
secx=1/2
cosx=2 no solution
tan (pi/2-x)
cotx (use equation)
reference <
degrees to x
tanx=1/3; in QIII what is sinx
draw triangle
1 to 1 if...
invertable
where does f=2(3^x) and g=3*(1/2)^x intersect?
ln(2*3^x)=ln(3*1/2)^x ln2+xln3=ln3+xln1/2 put x to same side and solve
solve for x 3^(2x+1)=7
log bae subtract 1, divide by 2
change base loga(b)=
logc(b)/logc(a)
sin(arcsinpi)
no sol
arccos0= cos0= cospi/2=
pi/2 1 0
vertical asymptotes of y=tanx
pi/2 +or-npi
angular speed of minute hand linear speed of minute hand (r=14) how far does tip move in 30min?
pi/30 pi/30*14 pi/30*14*30
Lax= 50,000 ppl in 2000 growth r8= 4% whats pop in 2008? 1993?
q= 50,000e^.04*8 q= 50,000e^.04*-7
whats growth rate if 5000 principle compounds continuously. doubles in 5yrs whats value in another 9yrs?
r=ln2/5 5000e^(14ln2/5)
secx((tanx^2)(-3))=0 solve for x
secx=0--> 1/cosx=0; no solution tan^2x=3; x=pi/3+npi
cos -15pi/4 --> (3 3/4)
sqrt2/2
tan7pi/3 -->(2 1/3pi)
sqrt3
sinx+cosx=2
square each side no solution because sin/cos are less than 1
|a|= magnitude
square terms and square root (hypotenuse)
tugboat1= 3000 lb force, direction N60W. Tug2= 2000lb. Which direction to make ship travel north?
t1=<-3000cos30, 3000cos30> t2= <2000cosx, 2000cosx> *add together; x value=0
cot3pi
undefined
sec 3pi/2
undefined
sinx=sin4
x=4+2npi x=3pi-4+2npi
sinx=1/3 solve for x
x=arcsin1/3+2npi x=pi-arcsin1/3+2npi
secx=-4 solve for x
x=pi-arcsec(4)+2npi x=pi+arcsec(4)+2npi
inverse of y=(x+1/2x-2)^1/3
y= 1+2x^3/2x^3-1
is <2,-4> perp to <6,3>?
yes
