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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

The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply

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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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