Law of Sines and Cosines
What is the Law of Sines and what triangles can this formula be used in?
sinA/a = sinB/b = sinC/c
SSA, ASA, and AAS
What does the parent function of an exponential look like?
Write the down all the critical points
f(x)=abx
horizontal asymptote at y=0
y-intercept at (0,1)
What is the parent function of a logarithmic graph?
*Write down all critical points
f(x)=alogb(x)
vertical asymptote: x=0
x-intercept at (1,0)
What happens to the ordered pair (a,b) in an inverse function?
(a,b)→(b,a)
What are the six trig functions and their graphs?
f(x)=cos(x)
f(x)=sin(x)
f(x)=tan(x)
f(x)=sec(x)
f(x)=csc(x)
f(x)=cot(x)
What is the Law of Cosine and what triangles does this formula use?
a2 = b2 + c2 - 2bc(cosA)
SAS and SSS
What is the relationship between an exponential and logarithmic function?
They are opposites of each other.
What is the change of base formula?
logbx= log(x)/log(b)= ln(x)/ln(b)
What's the test used to see if the equation is actually an inverse function?
Horizontal Line Test
Graph and analyze the following function:
g(x)=2sec2(x+pi)-1
Transformation:
y= stretch of 2, down one
x=shrink of 1/2, left pi
Period: pi
Domain: {x|xER, x≠ -pi/4 + pi/2n}
Range: {y|yER, y≤-3 ∪ y≥1}
Triangle ABC has side lengths of 5 and 3, with an angle of 40°.
Find the missing side length by either using Law of Sines or Law of Cosines.
c2=32+52-2(3)(5)cos40
c=√9+25-(30)(.766044)
c=3.319
The population of Smallville in the year 1890 was 6250. Assume the population increased at a rate of 2.75% per year.
a.) Estimate the population in 1930 and 1945.
b.) Predict when the population reached 20,000.
P(t)=6250(1+.0275)t
P(t)= 6250(1.0275)t
a.) 1915=12315 ; 1940= 24265
b.) 50,000=(1.0275)t
t=76.65
The population will reach 20,000 in late 1966.
Write an equation and sketch the graph for the following logarithmic expression:
log32+log34 / log324-log38
log3(4 x 2) / log3(24/8)
((*Remember: (+) means to multiply, (-) means to divide))
y=3x-8
(The graph should have an x-intercept at (log38,0), a y-intercept at (0,-7), and a horizontal asymptote y=-8.)
Find the inverse of the following expression:
x=y/y+1
x=y/y+1
x(y+1)=y *multiply by y+1
xy+x=y *Distributive Property
xy-y=-x *isolate the y terms
y(x-1)=-x *factor out y
y= -x/x-1 *divide by x-1
y= x/1-x *multiply numerator and denominator by -1
Sketch a graph of the following (label your axes/important points).
f(x)= tan2(x)/sec(x)+1
f(x)= tan2(x)/sec(x)+1
(sec2(x)-1)(sec(x)+1)/)sec(x)+1
f(x)= sec(x)-1
A business office just recently built a flag pole by the office. The owner wants to know the height of the flag pole and the angle of depressions from the ground to the top of the building.
(*Draw the picture for the problem as reference.)
Find the height of the pole using the Law of Cosines.
1. a2= b2+c2-2bc(cosA)
2. a2= 2702+1352-2(270)(135)(cos14)
3. a2= 72900+18225-2(36450)(cos14)
4. a2=19125-72900(cos14)
a2=18224(cos14)
a2= 2492.035802
*square root both sides
a= 49.92029449
The height of the flag pole is 49.920 ft tall.
The half-life of a certain radioactive substance is 14 days. There are 6.6g present initially.
a.) Express the amount of substance remaining as a function of time t.
b.) When will there be less than 1g remaining?
*initial=6.6g
a.) P(t)=6.6eln(1/2(t))
b.) after 38.11 days
Graph and analyze the following:
f(x)=ln-1/2(x+5)-3
Transformations:
y= down 3 ; x= flip, stretch of 2, left 5
(1,0)→(-7,-3)
x=0→ x=-5
*you can't take the log of a negative
0=ln-1/2(x+5)-3 *add three to both sides
3=ln-1/2(x+5) *raise e to both sides
e3=eln-1/2(x+5) *e and ln cancel out
e3=-1/2(x+5)
-2e3-5=x *x-intercept*
Domain: {x|xER, x<5}
Range: {y|yER}
EB: the limit as x approaches negative infinity is infinity
the limit as x approaches -5 is negative infinity
What is the inverse composition rule?
1. f(g(x))=x for every x in the domain of g and
2. g(f(x))=x for very x in the domain of f
Graph the following to trig functions. Are they the same? Why or why not?
f(x)=sin(x-pi/2)
f(x)=-cos(x)
They are the same.
A business office just recently built a flag pole by the office. The owner wants to know the height of the flag pole and the angle of depressions from the ground to the top of the building.
(*Draw the picture for the problem as reference.)
Find the angle using the Law of Sines.
1. sinC/c=sinA/a
2. sinC/120=sin90/270 (*multiply 120 to get sinC alone)
3. sinC=((sin90)(120))/270)
4. (*use sine inverse) C=sin-1((sin90)(120))/270)
C= 26.28779996
The angle of depression is 26.388˚ from the top of the building to the bottom of the flag pole.
1. The following data represent the temperature T (Fahrenheit) in Kansas City, Missouri on March 12,2015. Let t represent the number of hours after midnight.
a.) Determine whether a linear or cubic regression would best model temperature. Give the equation (to three decimal places) of each model and explain why one model is better than the other.
b.) Use your model to predict the temperature at 5 pm.
c.) Write one sentence that interprets the y-intercept of the model in context.
a.) linear: f(x)=1.226x+39.696
b.) f(17)=60.5ºF
c.) At midnight the temperature will be at about 39.696ºF (or 39.7).
Graph and analyze the following piecewise function. (check your graph on a calculator when done)
f(x)={ln(-x)+4, x≤0}
{-secx+1,x≥0}
(1,0)→(-1,4)
(e,1)→(-e,5)
0=ln(-x)=4
-4=ln(-x)
Change ln to e.
e-4=-x
Divide by -1 to both sides
x=-e-4
{x|xER,x<3pi/2}
{y|yER}
Limits: as x approaches infinity, y approaches infinity
as x approaches 3pi/2, y approaches infinity
Verify the following inverse:
f(x)=x3+1 and g(x)= (3√x-1)
f(g(x))=f3√x-1=(3√x-1)3+1=x-1+1=x
g(f(x))= g(x3+1)= (3√(x3+1)-1) = 3√x3=x
The bottom of a ferris wheel is 3ft off the ground. The radius is 20 ft. You begin moving continuously at the top of the wheel and it takes you 6 minutes to complete on rotation.
a. Write an equation that models the height of the ground as a function of time (both sine and cosine).
b. At what time you 20 ft off the ground in the first rotation?
a.
f(x)=20cos(pi/3)x + 23
f(x)=-20sin((pi/3)(x-3/2)) + 23
b.
When t=1.634 and t=4.356 (minutes) are you at 20 ft.