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Suppose a body has a force of 3 pounds acting on it to the left, 4 pounds acting on it upward, and 2 pounds acting on it 30° from the horizontal. What single force is needed to produce a state of equilibrium on the body? Draw the vector.

5.1583 pounds, 75.8° from the horizontal

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Chapter review exercises

Non-right Triangles: Law of Sines

For the following exercises, assume α is opposite side a , β is opposite side b , and γ is opposite side c . Solve each triangle, if possible. Round each answer to the nearest tenth.

β = 50° , a = 105 , b = 45

Not possible

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α = 43.1° , a = 184.2 , b = 242.8

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Solve the triangle.

Triangle with standard labels. Angle A is 36 degrees with opposite side a unknown. Angle B is 24 degrees with opposite side b = 16. Angle C and side c are unknown.

C = 120° , a = 23.1 , c = 34.1

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Find the area of the triangle.

A triangle. One angle is 75 degrees with opposite side unknown. The adjacent sides to the 75 degree angle are 8 and 11.
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A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 2.1 km apart, to be 25° and 49°, as shown in [link] . Find the distance of the plane from point A and the elevation of the plane.

Diagram of a plane flying over a highway. It is to the left and above points A and B on the ground in that order. There is a horizontal line going through the plan parallel to the ground. The angle formed by the horizontal line, the plane, and the line from the plane to point B is 25 degrees. The angle formed by the horizontal line, the plane, and point A is 49 degrees.

distance of the plane from point A : 2.2 km, elevation of the plane: 1.6 km

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Non-right Triangles: Law of Cosines

Solve the triangle, rounding to the nearest tenth, assuming α is opposite side a , β is opposite side b , and γ s opposite side c : a = 4 ,   b = 6 , c = 8.

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Solve the triangle in [link] , rounding to the nearest tenth.

A standardly labeled triangle. Angle A is 54 degrees with opposite side a unknown. Angle B is unknown with opposite side b=15. Angle C is unknown with opposite side C=13.

B = 71.0° , C = 55.0° , a = 12.8

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Find the area of a triangle with sides of length 8.3, 6.6, and 9.1.

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To find the distance between two cities, a satellite calculates the distances and angle shown in [link] (not to scale). Find the distance between the cities. Round answers to the nearest tenth.

Diagram of a satellite above and to the right of two cities. The distance from the satellite to the closer city is 210 km. The distance from the satellite to the further city is 250 km. The angle formed by the closer city, the satellite, and the other city is 1.8 degrees.

40.6 km

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Polar Coordinates

Plot the point with polar coordinates ( 3 , π 6 ) .

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Plot the point with polar coordinates ( 5 , 2 π 3 )


Polar coordinate grid with a point plotted on the fifth concentric circle 2/3 the way between pi and 3pi/2 (closer to 3pi/2).

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Convert ( 6 , 3 π 4 ) to rectangular coordinates.

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Convert ( 2 , 3 π 2 ) to rectangular coordinates.

( 0 , 2 )

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Convert ( 7 , 2 ) to polar coordinates.

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Convert ( 9 , 4 ) to polar coordinates.

( 9.8489 , 203.96° )

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For the following exercises, convert the given Cartesian equation to a polar equation.

For the following exercises, convert the given polar equation to a Cartesian equation.

r = 7 cos θ

x 2 + y 2 = 7 x

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r = 2 4 cos θ + sin θ

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For the following exercises, convert to rectangular form and graph.

Polar Coordinates: Graphs

For the following exercises, test each equation for symmetry.

r = 4 + 4 sin θ

symmetric with respect to the line θ = π 2

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Sketch a graph of the polar equation r = 1 5 sin θ . Label the axis intercepts.


Graph of the given polar equation - an inner loop limaçon.

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Sketch a graph of the polar equation r = 5 sin ( 7 θ ) .

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Sketch a graph of the polar equation r = 3 3 cos θ


Graph of the given polar equation - a cardioid.

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Polar Form of Complex Numbers

For the following exercises, find the absolute value of each complex number.

Write the complex number in polar form.

1 2 3 2 i

cis ( π 3 )

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For the following exercises, convert the complex number from polar to rectangular form.

z = 3 cis ( 40° )

2.3 + 1.9 i

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For the following exercises, find the product z 1 z 2 in polar form.

z 1 = 2 cis ( 89° )

z 2 = 5 cis ( 23° )

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z 1 = 10 cis ( π 6 )

z 2 = 6 cis ( π 3 )

60 cis ( π 2 )

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For the following exercises, find the quotient z 1 z 2 in polar form.

z 1 = 12 cis ( 55° )

z 2 = 3 cis ( 18° )

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z 1 = 27 cis ( 5 π 3 )

z 2 = 9 cis ( π 3 )

3 cis ( 4 π 3 )

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For the following exercises, find the powers of each complex number in polar form.

Find z 4 when z = 2 cis ( 70° )

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Find z 2 when z = 5 cis ( 3 π 4 )

25 cis ( 3 π 2 )

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Questions & Answers

I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
Carlos Reply
How can you tell what type of parent function a graph is ?
Mary Reply
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
Karim Reply
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
GREAT ANSWER THOUGH!!!
Darius
Thanks.
Thomas
Â
Thomas
It is the  that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
unknown Reply
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
Ef Reply
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
KARMEL Reply
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
Rima Reply
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
Brittany Reply
how do you find the period of a sine graph
Imani Reply
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
Jhon Reply
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
Baptiste Reply
the sum of any two linear polynomial is what
Esther Reply
divide simplify each answer 3/2÷5/4
Momo Reply
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
austin Reply
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
6000
Robert
more than 6000
Robert
For Plan A to reach $27/month to surpass Plan B's $26.50 monthly payment, you'll need 3,000 texts which will cost an additional $10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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