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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

can you not take the square root of a negative number
Sharon Reply
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
Elaine Reply
can I get some pretty basic questions
Ama Reply
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
Amara Reply
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
Mars Reply
what is the domain of f(x)=x-4/x^2-2x-15 then
Conney Reply
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
jeric Reply
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
jeric Reply
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
Abena Reply
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
Ashley Reply
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Fiston Reply
Where do the rays point?
Spiro
x=-b+_Гb2-(4ac) ______________ 2a
Ahlicia Reply
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
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
Brad
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
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

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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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