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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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
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John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
what is inorganic
emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
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Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
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hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
Reofrir 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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