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For the following exercises, evaluate each root.

Evaluate the cube root of z when z = 64 cis ( 210° ) .

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Evaluate the square root of z when z = 25 cis ( 3 π 2 ) .

5 cis ( 3 π 4 ) , 5 cis ( 7 π 4 )

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For the following exercises, plot the complex number in the complex plane.

Parametric Equations

For the following exercises, eliminate the parameter t to rewrite the parametric equation as a Cartesian equation.

{ x ( t ) = 3 t 1 y ( t ) = t

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{ x ( t ) = cos t y ( t ) = 2 sin 2 t  

x 2 + 1 2 y = 1

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Parameterize (write a parametric equation for) each Cartesian equation by using x ( t ) = a cos t and y ( t ) = b sin t for x 2 25 + y 2 16 = 1.

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Parameterize the line from ( 2 , 3 ) to ( 4 , 7 ) so that the line is at ( 2 , 3 ) at t = 0 and ( 4 , 7 ) at t = 1.

{ x ( t ) = 2 + 6 t y ( t ) = 3 + 4 t

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Parametric Equations: Graphs

For the following exercises, make a table of values for each set of parametric equations, graph the equations, and include an orientation; then write the Cartesian equation.

{ x ( t ) = 3 t 2 y ( t ) = 2 t 1

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{ x ( t ) = e t y ( t ) = 2 e 5 t

y = 2 x 5

Plot of the given parametric equations.
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{ x ( t ) = 3 cos t y ( t ) = 2 sin t

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A ball is launched with an initial velocity of 80 feet per second at an angle of 40° to the horizontal. The ball is released at a height of 4 feet above the ground.

  1. Find the parametric equations to model the path of the ball.
  2. Where is the ball after 3 seconds?
  3. How long is the ball in the air?
  1. { x ( t ) = ( 80 cos ( 40° ) ) t y ( t ) = 16 t 2 + ( 80 sin ( 40° ) ) t + 4
  2. The ball is 14 feet high and 184 feet from where it was launched.
  3. 3.3 seconds
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Vectors

For the following exercises, determine whether the two vectors, u and v , are equal, where u has an initial point P 1 and a terminal point P 2 , and v has an initial point P 3 and a terminal point P 4 .

P 1 = ( 1 , 4 ) , P 2 = ( 3 , 1 ) , P 3 = ( 5 , 5 ) and P 4 = ( 9 , 2 )

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P 1 = ( 6 , 11 ) , P 2 = ( 2 , 8 ) , P 3 = ( 0 , 1 ) and P 4 = ( 8 , 2 )

not equal

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For the following exercises, use the vectors u = 2 i j , v = 4 i 3 j , and w = 2 i + 5 j to evaluate the expression.

For the following exercises, find a unit vector in the same direction as the given vector.

b = −3 i j

3 10 10 i 10 10 j

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For the following exercises, find the magnitude and direction of the vector.

−3 , −3

Magnitude: 3 2 , Direction: 225°

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For the following exercises, calculate u v .

u = −2 i + j and v = 3 i + 7 j

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u = i + 4 j and v = 4 i + 3 j

16

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Given v = −3 , 4 draw v , 2 v , and 1 2 v .

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Given the vectors shown in [link] , sketch u + v , u v and 3 v .

Diagram of vectors v, 2v, and 1/2 v. The 2v vector is in the same direction as v but has twice the magnitude. The 1/2 v vector is in the same direction as v but has half the magnitude.


Diagram of vectors u and v. Taking u's starting point as the origin, u goes from the origin to (4,1), and v goes from (4,1) to (6,0).

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Given initial point P 1 = ( 3 , 2 ) and terminal point P 2 = ( 5 , 1 ) , write the vector v in terms of i and j . Draw the points and the vector on the graph.

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Practice test

Assume α is opposite side a , β is opposite side b , and γ is opposite side c . Solve the triangle, if possible, and round each answer to the nearest tenth, given β = 68° , b = 21 , c = 16.

α = 67.1° , γ = 44.9° , a = 20.9

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Find the area of the triangle in [link] . Round each answer to the nearest tenth.

A triangle. One angle is 60 degrees with opposite side 6.25. The other two sides are 5 and 7.
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A pilot flies in a straight path for 2 hours. He then makes a course correction, heading 15° to the right of his original course, and flies 1 hour in the new direction. If he maintains a constant speed of 575 miles per hour, how far is he from his starting position?

1712 miles

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Convert ( 2 , 2 ) to polar coordinates, and then plot the point.

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Convert ( 2 , π 3 ) to rectangular coordinates.

( 1 , 3 )

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Convert the polar equation to a Cartesian equation: x 2 + y 2 = 5 y.

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Convert to rectangular form and graph: r = 3 csc θ .

y = 3

Plot of the given equation in rectangular form - line y=-3.
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Test the equation for symmetry: r = 4 sin ( 2 θ ).

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Graph r = 3 + 3 cos θ .


Graph of the given equations - a cardioid.

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Graph r = 3 5 sin θ .

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Find the absolute value of the complex number 5 9 i .

106

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Write the complex number in polar form: 4 + i .

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Convert the complex number from polar to rectangular form: z = 5 cis ( 2 π 3 ) .

5 2 + i 5 3 2

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Given z 1 = 8 cis ( 36° ) and z 2 = 2 cis ( 15° ) , evaluate each expression.

z 1

2 2 cis ( 18° ) , 2 2 cis ( 198° )

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Plot the complex number −5 i in the complex plane.

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Eliminate the parameter t to rewrite the following parametric equations as a Cartesian equation: { x ( t ) = t + 1 y ( t ) = 2 t 2 .

y = 2 ( x 1 ) 2

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Parameterize (write a parametric equation for) the following Cartesian equation by using x ( t ) = a cos t and y ( t ) = b sin t : x 2 36 + y 2 100 = 1.

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Graph the set of parametric equations and find the Cartesian equation: { x ( t ) = 2 sin t y ( t ) = 5 cos t .


Graph of the given equations - a vertical ellipse.

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A ball is launched with an initial velocity of 95 feet per second at an angle of 52° to the horizontal. The ball is released at a height of 3.5 feet above the ground.

  1. Find the parametric equations to model the path of the ball.
  2. Where is the ball after 2 seconds?
  3. How long is the ball in the air?
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For the following exercises, use the vectors u = i − 3 j and v = 2 i + 3 j .

Find 2 u − 3 v .

−4 i − 15 j

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Find a unit vector in the same direction as v .

2 13 13 i + 3 13 13 j

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Given vector v has an initial point P 1 = ( 2 , 2 ) and terminal point P 2 = ( 1 , 0 ) , write the vector v in terms of i and j . On the graph, draw v , and v .

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Questions & Answers

how does Neisseria cause meningitis
Nyibol Reply
what is microbiologist
Muhammad Reply
what is errata
Muhammad
is the branch of biology that deals with the study of microorganisms.
Ntefuni Reply
What is microbiology
Mercy Reply
studies of microbes
Louisiaste
when we takee the specimen which lumbar,spin,
Ziyad Reply
How bacteria create energy to survive?
Muhamad Reply
Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments
_Adnan
But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?
Muhamad
they make spores
Louisiaste
what is sporadic nd endemic, epidemic
Aminu Reply
the significance of food webs for disease transmission
Abreham
food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.
Mark
explain assimilatory nitrate reduction
Esinniobiwa Reply
Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).
Elkana
This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products
Elkana
Examples of thermophilic organisms
Shu Reply
Give Examples of thermophilic organisms
Shu
advantages of normal Flora to the host
Micheal Reply
Prevent foreign microbes to the host
Abubakar
they provide healthier benefits to their hosts
ayesha
They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!
Mark
what is cell
faisal Reply
cell is the smallest unit of life
Fauziya
cell is the smallest unit of life
Akanni
ok
Innocent
cell is the structural and functional unit of life
Hasan
is the fundamental units of Life
Musa
what are emergency diseases
Micheal Reply
There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️
_Adnan
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Innocent
I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life
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Heyy Lubega hussein where are u from?
_Adnan
en français
Adama
which site have a normal flora
ESTHER Reply
Many sites of the body have it Skin Nasal cavity Oral cavity Gastro intestinal tract
Safaa
skin
Asiina
skin,Oral,Nasal,GIt
Sadik
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all
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by fussion
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Micheal
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part of a tissue or an organ being wounded or bruised.
Wilfred
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Binomial nomenclature
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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