unit 10 precalc apex
find the product of the complex numbers. express ur answer in trigonometric form z1 = 3(cos(π/4) + isin(π/4)) z2 = 7(cos(3π/8) + isin(3π/8))
21(cos(5π/8) + isin(5π/8))
Which of the following is a third root of the given complex number? 27(cos(pi/5)+isin(pi/5))
3(cos(pi/15)+i sin(pi/15)) 3(cos(11pi/15)+i sin(11pi/15))
convert the complex number 2-2sqrt3 i into its polar representation
4(cos(300)+isin(300))
Convert the following complex number into its polar representation: 2sqrt 3+2i
4(cos(30°) + i*sin(30°))
find the quotient, write your answer in standard form: (3-i)/(3+i)
4/5-3/5i
rewrite the square root expression as an imaginary number: sqrt -25
5i
A complex number will in general have ____ sixth complex roots
6
write the product in standard form: (2-3i)(2+i)
7 - 4i
find the conjugate of the complex number: 7-3i
7+3i
simplify: (-2+3i)+2(5+6i)
8+15i
Which features are present in this polar graph? (cardiod on right side of graph)) A.Symmetry about the line theta =pi/2 B.Symmetry about the polar axis theta= 0 C.Symmetry about the pole (0,0) D.No symmetry
Symmetry about the polar axis theta= 0
The displacement of a mass suspended on a spring is modeled by the function y = 3sin(4)(pi)(t), where y is measured in inches and t in seconds. The amplitude, period, and frequency of the motion of the mass are given by
amplitude=2 period=2/3 frequency=3/2
what is the name of the shape graphed by the function r=4sin theta
circle
(TRUE or FALSE) When cubing a complex number using De Moivre's theorem, there are three answers.
false
(True or False) The rectangular coordinates of a point are not unique
false
(true or false) A rectangular equation that is a function will be a function in any coordinate system
false
(true or false) Let f(x) = Tan^-1 x. Then the domain of f(x) is all real numbers and the range (0, pi)
false
what is the name of the shape graphed by the function: theta=-4pi/3
line
When a complex number z is written in its polar form,z = r(cos theta + i sin theta) , the nonnegative number r is called the ____________, or modulus, of z.
magnitude
When a complex number z is written in its polar form,z = r(cos theta + i sin theta) , the nonnegative number r is called the ____________ of z.
modulus
An equation whose variables are polar coordinates is called a(n) _______________
polar equation
In polar coordinates, the origin is called the _____.
pole
(true or false) The value of r represents the directed distance when plotting a point in polar coordinates
true
An equation whose ____ are polar coordinates is called a polar equation
variables
find all solutions to the equation secx(2cosx-sqrt2)=0
x=pi/4+2kpi and 7pi/4+2kpi, where k is any positive integer
transform the following polar equation into an equation in rectangular coordinates: r=4
x^2+y^2=16
which rectangular equation is equivalent to the polar equation rsin theta=-2
y=-2
Which of the following expressions is a sixth root of unity? 1. i 2. -1 3. -i 4. All of the above are correct
-1
find the exact value of csc(sin^-1(-1)
-1
If v1= (3, -4) and v2= (2, 6), then v1*v2 is equal to which of the following?
-18
Another representation in polar coordinates of the point is (2,pi/6) is(---,7pi/6)
-2
which set of polar coordinates are plotted in the graph below
-2, 2pi/3
A skyscraper casts a shadow 200 feet long. If the angle of elevation of the sun is 49 , then the height of the skyscraper is _____.
180.08 feet
simplify: (4-3i)+(2+6i)
6+3i
the function f(x)=cot(3x-pi/2) has
Period: pi/3 asymptote: x=pi/6
what is the name of the shape graphed by the function r=1+sintheta
cardiod
convert 4(cos(240 degrees) +isin 240 degrees)) into rectangular form
(-2,-2sqrt3)
Which set of rectangular coordinates describes the same location as the polar coordinates (-2 2pi)
(-2,0)
Which set of rectangular coordinates describes the same location as the polar coordinates 3sqrt2 5pi/4
(-3,3)
convert the polar representation of this complex number into its regular form: z=5(cospi+isinpi)
(-5,0)
Which set of polar coordinates describes the same location as the rectangular coordinates (-5 0)
(5, 180 degrees)
which set of polar coordinates names the same point as (-5,pi/4)
(5, 5π/4)
convert (5,pi/4) to rectangular coordinates
(5√2/2,5√2/2)
Use de moivre's theorem to write the complex number in trigonometric form (cos(2pi/7)+isin(2pi/7))^5
(cos(10pi/7)+isin(10pi/7))^5
Use de moivre's theorem to write the complex number in trigonometric form (cos(3pi/5)+isin(3pi/5))^3
(cos(9pi/5)+isin(9pi/5))
find the exact value of cos(7pi/8)
-sqrt(2+sqrt2)/2
which complex number's graph is shown below (point at 2, -2)
2sqrt(2)(cos(7pi )/(4)+isin(7pi )/(4))
simplify: (5+2i)-(-3+4i)
8-2i
the function f(x)=-2cos(3x-(pi )/(4)) has
amplitude 2 and period 2pi/3
Which of the following pairs of coordinates do not represent the same point in polar coordinates? Choose all that apply a. (r , theta) and (r , theta+2pi) b. (r , theta) and (-r , theta+2pi) c. (r , theta) and (-r , theta) d. (r , theta) and (-r , theta+pi)
b. (r , theta) and (-r , theta+2pi) c. (r , theta) and (-r , theta)
If triangle ABC has angle 52 side a=200 and side b=234 then which of the following is true
b=67.2 or 112.8
what is the name of the shape graphed by the function r=3+3costheta
cardiod
(true or false) Let f(x) = cos-1x. Then the domain of f(x) is all real number and the range is [-1,1]
false
(true or false) One disadvantage of polar equations is that you can use a polar function to describe a graph whose equation in rectangular coordinates would not be considered a function
false
(true or false) The motion of a pendulum for which the maximum displacement from equilibrium varies is an example of simple harmonic motion
false
(true or false) As long as its argument is restricted to [0,2pi] the polar form of a complex number is unique.
true
find the exact value of sin(cos^-1(3/7))
2 sqrt10/7
what is the name of the shape graphed by the function theta=-pi/6
line
Which set of polar coordinates describes the same location as the rectangular coordinates (1,-1)
sqrt 2, 315 degrees
which pair represents the same complex number
sqrt(2)(cos(-(pi )/(4))+isin(-(pi )/(4)))
calculate: |-3+4i|
5
An arc of length 10 meters is formed by a central angle A on a circle of radius 4. The measure of A in degrees (to two decimal places) is _____. A. 143.24 B. 2.50 C. 216.76 D. 36.76
A. 143.24
What is true about the complex number -5 - 5i? Check all that apply. A. The modulus is 10. B. It lies in quadrant 4. C. It lies in quadrant 2. D. The modulus is 5 square root of 2.
B. The modulus is 5sqrt2 C. It lies in quadrant 4
In this triangle, which of the following is true?
a. The measure of x is 55° and a ≈ 11.47
assume the graph of a function of the form y=acos(k(x-b)) is given below. which of the following are possible values for a,k,b
a=4, k=2,b=-pi/4
what is the name of the shape graphed by the function r=1+sin theta
cardioid
(True or false) The value of r represents the reference angle when plotting a point in polar coordinates
false
In polar coordinates, the positive x-axis is called the ___ axis
polar
which of the following are not trigonometric identities check all that apply
sec^2x+csc^2x=1 tan^2x+sec^2x=1
(true or false) De Moivre's theorem is used to raise a complex number to a positive integer power
true
(true or false) In (r, theta), the value of r can be negative.
true
the graph below represents which of the following functions
y=3cos(2(x+(pi )/(3)))
which expression is a cube root of -1+isqrt 3
3sqrt2 (cos(40 degrees) + isin(40 degrees))
the following is the graph of (graph with lines that look likes stretched 5's)
y=tan(2(x-(pi )/(6)))
convert the polar representation of this complex number into its regular form: z=4(cos135 degrees+isin135 degrees)
-2sqrt2, 2 sqrt2
which of the following are solutions to the equation tan^2(2x)-1=0
-pi/8 3pi/8
What value in the nth root of unity for any value n?
1
Find the product of the complex numbers. Express your answer in trigonometric form: z1 = 7(cos15° + isin15°) z2 = 2(cos110° + isin110°)
14(cos125° + isin125°)
Find the product of the complex numbers. Express your answer in trigonometric form: z1 = 7(cos15° + isin15°) z2 = 2(cos110° + isin110°)
14(cos125° + isin125°)
If v1=(2,5) and v2=(4,-3), then the angle between the two vectors is____degrees. Round your answer to two decimal places
31.33
convert the polar representation of this complex number into its standard form: z=4(cos 5pi/6+isin5pi/6
-2sqrt3 +2i
convert the polar representation of this complex number into its standard form: 8(cos(4pi/3)+isin(4pi/3))
-4-4sqrt3i
which of the following are solutions to the equation: 2cos^2x-1=0
-7pi/4,3pi/4,15pi/4
Swanson plans to attempt to swim across the lake from point A to point B as shown below. To determine how far he will have to swim, he has chosen a point C and determined the labeled distances. How many miles will he have to swim? Round your answer to two decimal places
70.40
Use de moivre's theorem to write the complex number in trigonometric form: (1-isqrt3)^3
8(cospi+isinpi)
find all solutions to the equation cos x cos(3x)+sin x sin(3x)=0 on the interval (0,2pi)
pi/4,3pi/4,5pi/4,7pi/4
the graph shown below represents which of the following polar equations
r=3-3sin theta
which equation will produce the graph shown (infinity in middle of graph)
r^2=9cos(2 theta)
which expression is a cube root of -2i
3sqrt2 (cos(210 degrees)+isin(210 degrees))
what is the name of the shape described in the graph (diagonal infinity sign)
lemniscate
what is the name of the shape described below (looks like circle with a smaller circle hugging the edge)
limacon
Convert the following complex number into its polar representation: -4i
4(cos 3pi/2 + isin 3pi/2)
Use De Moivre's theorem to write the complex number in standard form. ( sqrt2 (cos(10 degrees)+isin(10 degrees))^6
4+4i*sqrt(3)
Find the quotient of z1 by z2. Express your answer in trigonometric form: z1 = 4(cos(π/3) + isin(π/3)) z2 = 3(cos(2π/5) + isin(2π/5))
4/3 (cos(-π/15) + isin(-π/15))
if z1=5+5i and z2=8(cos(3pi/7)+isin(3pi/7)) then z1 z2=____? A. 8sqrt5(cos(19pi/28)+isin(19pi/28)) B. 40sqrt2(cos(3pi/28)+isin(3pi/28)) C. 40(cos(19pi/28)+isin(19pi/28)) D. 40sqrt2(cos(19pi/28)+isin+(19pi/28))
40sqrt2(cos(19pi/28)+isin+(19pi/28))
Find the quotient of the complex numbers. Express your answer in trigonometric form. z1= 5(cos25Deg + i sin25Deg) z2= 2(cos80Deg + isin80Deg)
5/2(cos305Deg-i sin305Deg)
Which of the following are correct statements concerning the roots of unity? Check all that apply. A. nth roots of unity are evenly spaced on the unit circle. B. 1 is always the nth root of unity. C. The number of nth roots of unity is n. D. -1 is always the nth root of unity.
A. nth roots of unity are evenly spaced on the unit circle. B. 1 is always the nth root of unity. C. The number of nth roots of unity is n.
When a complex number is written in its polar form, z=r(cos theta + i sin theta) , the angle theta is called the "argument" of _____.
z
Which of the following are identities? Check all that apply.
{secx-cosx}/{secx}=sin^2x tanxcosxcscx=1 1-tantany=cos(x-y)/cosxcoy
convert the polar representation of this complex number into its standard form: z=8(cos240 degrees+isin240 degrees)
-4-4sqrt3i