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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

difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
Feemark Reply
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

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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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