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mathematics
precalculus

Questions and Answers of Precalculus

For any triangle, derive the Law of Tangents. а -в) tan а — Ь (A – B) a + b (A + B) tan
For any triangle, derive the formulaa = b cos C + c cos BUse the fact that sin A = sin(180° - B - C).
Another form of Mollweide’s Formula isDerive it. sin a - b (A – B) cos CoS
For any triangle, Mollweide’s Formula (named after Karl Mollweide, 1774–1825) states thatDerive it. (A – B) cCos a + b sinGc)
George Washington Gale Ferris, Jr., designed the original Ferris wheel for the 1893 World’s Columbian Exposition in Chicago, Illinois. The wheel had 36 equally spaced cars each the size of a school
The distance from the Sun to Earth is approximately 149,600,000 km. The distance from the Sun to Venus is approximately 108,200,000 km.The elongation angle α is the angle formed between the line of
The distance from the Sun to Earth is approximately 149,600,000 kilometers (km). The distance from the Sun to Mercury is approximately 57,910,000 km. The elongation angle α is the angle formed
Two sensors are spaced 700 feet apart along the approach to a small airport. When an aircraft is nearing the airport, the angle of elevation from the first sensor to the aircraft is 20°, and from
One of the original Seven Wonders of the World, the Great Pyramid of Cheops was built about 2580 BC. Its original height was 480 feet 11 inches, but owing to the loss of its topmost stones, it is now
A forest ranger is walking on a path inclined at 5° to the horizontal directly toward a 100-foot-tall fire observation tower. The angle of elevation from the path to the top of the tower is 40°.
An awning that covers a sliding glass door that is 88 inches tall forms an angle of 50° with the wall. The purpose of the awning is to prevent sunlight from entering the house when the angle of
The navigator of a ship at sea spots two lighthouses that she knows to be 3 miles apart along a straight seashore. She determines that the angles formed between two line-of-sight observations of the
U.S. 41, a highway whose primary directions are north–south, is being constructed along the west coast of Florida.Near Naples, a bay obstructs the straight path of the road. Since the cost of a
On a certain automobile, the crankshaft is 3 inches long and the connecting rod is 9 inches long (see the figure).At the time when ∠OPQ is 15°, how far is the piston (P) from the center (O) of the
The famous Leaning Tower of Pisa was originally 184.5 feet high.* At a distance of 123 feet from the base of the tower, the angle of elevation to the top of the tower is found to be 60°. Find ∠RPQ
In attempting to fly from city P to city Q, an aircraft followed a course that was 10° in error, as indicated in the figure.After flying a distance of 50 miles, the pilot corrected the course by
Adam must fly home to St. Louis from a business meeting in Oklahoma City. One flight option flies directly to St. Louis, a distance of about 461.1 miles.A second flight option flies first to Kansas
A loading ramp 10 feet long that makes an angle of 18° with the horizontal is to be replaced by one that makes an angle of 12° with the horizontal. How long is the new ramp?
Pat needs to determine the height of a tree before cutting it down to be sure that it will not fall on a nearby fence. The angle of elevation of the tree from one position on a flat path from the
The highest bridge in the world is the bridge over the Royal Gorge of the Arkansas River in Colorado. Sightings to the same point at water level directly under the bridge are taken from each side of
An aircraft is spotted by two observers who are 1000 feet apart.As the airplane passes over the line joining them, each observer takes a sighting of the angle of elevation to the plane, as indicated
Use the illustration in Problem 39 to find the height QD of the mountain.Data from problem 39Consult the figure.To find the length of the span of a proposed ski lift from P to Q, a surveyor measures
Consult the figure.To find the length of the span of a proposed ski lift from P to Q, a surveyor measures ∠DPQ to be 25° and then walks off a distance of 1000 feet to R and measures ∠PRQ to be
At exactly the same time, Tom and Alice measured the angle of elevation to the moon while standing exactly 300 km apart. The angle of elevation to the moon for Tom was 49.8974° and the angle of
Coast Guard Station Able is located 150 miles due south of Station Baker.A ship at sea sends an SOS call that is received by each station. The call to Station Able indicates that the ship is located
Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results.b = 4, c = 5, B = 40°
Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results.a = 2, c = 1, C = 25°
Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results.b = 4, c = 5, B = 95°
Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results.a = 2, c = 1, C = 100°
Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results.a = 3, b = 7, A = 70°
Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results.b = 4, c = 6, B = 20°
Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results.b = 2, c = 3, B = 40°
Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results.a = 4, b = 5, A = 60°
Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results.a = 2, c = 1, A = 120°
Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results.b = 5, c = 3, B = 100°
Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results.b = 4, c = 3, B = 40°
Two sides and an angle are given. Determine whether the given information results in one triangle, two triangles, or no triangle at all. Solve any triangle(s) that results.a = 3, b = 2, A = 50°
Solve the triangle.B = 70°, C = 10°, b = 5
Solve the triangle.B = 20°, C = 70°, a = 1
Solve the triangle.A = 40°, B = 40°, c = 2
Solve the triangle.B = 10°, C = 100°, b = 2
Solve the triangle.A = 110°, C = 30°, c = 3
Solve the triangle.A = 70°, B = 60°, c = 4
Solve the triangle.A = 50°, C = 20°, a = 3
Solve the triangle.A = 40°, B = 20°, a = 2
Solve the triangle. a 30° 100
Solve the triangle. a 100 2 40°
Solve the triangle. 5°. 10° 5
Solve the triangle. a 45° 40°
Solve the triangle. 125 10 30°
Solve the triangle. 85 a 3 50°
Solve the triangle. a 40° 45° 4.
Solve the triangle. 95 a 45° 5
Triangles for which two sides and the angle opposite one of them are known (SSA) are referred to as the __________ ______.
True or FalseThe Law of Sines can be used to solve triangles where three sides are known.
True or FalseAn oblique triangle in which two sides and an angle are given always results in at least one triangle.
For a triangle with sides a, b, c and opposite angles A, B, C, the Law of Sines states that ________.
If none of the angles of a triangle is a right angle, the triangle is called ________.
The two triangles shown are similar. Find the missing length. 3 2.
If θ is an acute angle, solve the equation cos θ = √3/2.
The difference formula for the sine function sin(A - B) = ____.
The Gibb’s Hill Lighthouse. Southampton, Bermuda In operation since 1846, the Gibb’s Hill Lighthouse stands 117 feet high on a hill 245 feet high, so its beam of light is 362 feet above sea
The Freedom Tower is to be the centerpiece of the rebuilding of the World Trade Center in New York City. The tower will be 1776 feet tall (not including a broadcast antenna). The angle of elevation
A pool player located at X wants to shoot the white ball off the top cushion and hit the redball dead center. He knows from physics that the white ball will come off a cushion at the same angle as it
A surveillance satellite circles Earth at a height of h miles above the surface.Suppose that d is the distance, in miles, on the surface of Earth that can be observed from the satellite.See the
Find the value of the angle in degrees rounded to the nearest tenth of a degree. 2
The eyes of a basketball player are 6 feet above the floor. The player is at the free-throw line, which is 15 feet from the center of the basket rim (see the figure). What is the angle of elevation
A carpenter is preparing to put a roof on a garage that is 20 feet by 40 feet by 20 feet. A steel support beam 46 feet in length is positioned in the center of the garage. To support the roof,
A tourist at the top of the Gateway Arch (height, 630 feet) in St. Louis, Missouri, observes a boat moored on the Illinois side of the Mississippi River 2070 feet directly across from the Arch. She
The angle of inclination from the base of the John Hancock Center to the top of the main structure of the Willis Tower is approximately 10.3°. If the main structure of the Willis Tower is 1451 feet
The CNTower, located in Toronto, Canada, is the tallest structure in the Americas. While visiting Toronto, a tourist wondered what the height of the tower above the top of the Sky Pod is. While
To measure the height of Lincoln’s caricature on Mt. Rushmore, two sightings 800 feet from the base of the mountain are taken. If the angle of elevation to the bottom of Lincoln’s face is 32°
While taking a ride in a hot-air balloon in Napa Valley, Francisco wonders how high he is. To find out, he chooses a landmark that is to the east of the balloon and measures the angle of depression
A blimp, suspended in the air at a height of 500 feet, lies directly over a line from Soldier Field to the Adler Planetarium on Lake Michigan (see the figure). If the angle of depression from the
A camera is mounted on a tripod 4 feet high at a distance of 10 feet from George, who is 6 feet tall.See the illustration. If the camera lens has angles of depression and elevation of 20°, will
A highway whose primary directions are north–south is being constructed along the west coast of Florida.Near Naples, a bay obstructs the straight path of the road.Since the cost of a bridge is
Willis Tower in Chicago is the third tallest building in the world and is topped by a high antenna.A surveyor on the ground makes the following measurement:1.The angle of elevation from his position
Situated between Portage Road and the Niagara Parkway directly across from the Canadian Horseshoe Falls, the Falls Incline Railway is a funicular that carries passengers up an embankment to Table
A ship leaves the port of Miami with a bearing of S80°E and a speed of 15 knots. After 1 hour, the ship turns 90° toward the south.After 2 hours, maintaining the same speed, what is the bearing to
ADC-9 aircraft leaves Midway Airport from runway 4 RIGHT, whose bearing is N40°E.After flying for 1/2 mile, the pilot requests permission to turn 90° and head toward the southeast. The permission
A straight trail with an inclination of 17° leads from a hotel at an elevation of 9000 feet to a mountain lake at an elevation of 11,200 feet. What is the length of the trail?
The angle of elevation of the Sun is 35.1° at the instant the shadow cast by the Washington Monument is 789 feet long. Use this information to calculate the height of the monument.
Cygni, sometimes called Bessel’s Star (after Friedrich Bessel, who measured the distance from Earth to the star in 1838), is a star in the constellation Cygnus.(a) 61 Cygni is 11.14 light-years
One method of measuring the distance from Earth to a star is the parallax method. The idea behind computing this distance is to measure the angle formed between the Earth and the star at two
A security camera in a neighborhood bank is mounted on a wall 9 feet above the floor. What angle of depression should be used if the camera is to be directed to a spot 6 feet above the floor and 12
A state trooper is hidden 30 feet from a highway. One second after a truck passes, the angle between the highway and the line of observation from the patrol car to the truck is measured. See the
A laser beam is to be directed through a small hole in the center of a circle of radius 10 feet. The origin of the beam is 35 feet from the circle (see the figure). At what angle of elevation should
At 10 AM on April 26, 2009, a building 300 feet high casts a shadow 50 feet long. What was the angle of elevation of the Sun?
A 22-foot extension ladder leaning against a building makes a 70° angle with the ground. How far up the building does the ladder touch?
Suppose that you are headed toward a plateau 50 meters high. If the angle of elevation to the top of the plateau is 20°, how far are you from the base of the plateau?
A person in a small boat, offshore from a vertical cliff known to be 100 feet in height, takes a sighting of the top of the cliff. If the angle of elevation is found to be 25°, how far offshore is
The tallest tower built before the era of television masts, the Eiffel Tower was completed on March 31, 1889. Find the height of the Eiffel Tower (before a television mast was added to the top) using
Find the distance from A to C across the pond illustrated in the figure. 40° 100 ft
Find the distance from A to C across the gorge illustrated in the figure. 35° C- 100 ft
Aright triangle contains an angle of radian.(a) If one leg is of length 3 meters, what is the length of the hypotenuse?(b) There are two answers. How is this possible?
A right triangle contains a 25° angle.(a) If one leg is of length 5 inches, what is the length of the hypotenuse?(b) There are two answers. How is this possible?
Aright triangle has a hypotenuse of length 10 centimeters. If one angle is 40°, find the length of each leg.
A right triangle has a hypotenuse of length 8 inches. If one angle is 35°, find the length of each leg.
The hypotenuse of a right triangle is 3 feet. If one leg is 1 foot, find the degree measure of each angle.
The hypotenuse of a right triangle is 5 inches. If one leg is 2 inches, find the degree measure of each angle.

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