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1. Find the elevation angle of the sun if a man of 1.75 m. tall, casts a shadow of 82 cm. length on the floor.

2. From a point which is 12 m. soil, an observer obtains a measurement of 53 degrees to the angle of depression of an object found in the soil. About how far is the object point on the ground which is directly under the observer?

3. The string of a comet is taut and forming an angle of 48 degrees with the horizontal. Find the height of the comet with respect the ground, if the line measures 87 m. and the end of the rope is held to 1.3 m. soil.

4. A plane flying at an altitude of 10,000 feet and passes directly over a fixed object on land. A minute later, the depression angle of the object is 42 degrees. Determine the approximate speed of the aircraft.

5. Calculate the width of the street, if an observer on a building, see the other side of it at an angle of 60 degrees from the horizontal.

6. One person is in the window of his apartment which is located at 8m. soil and see the building opposite. Top with an angle of 30 degrees and the bottom at an angle of depression 45 degrees. Determine the height of the building said.

7. A river has two parallel banks. Since the points P and Q from one side, there is a point R on the opposite bank. If the visual form to the address of the bank angle of 40 degrees and 50 degrees respectively, and the distance between points P and Q is 30 meters, determine the width of river.

8. A box is located on a wall so that its lower edge is at a distance of 20 cm. above the eye of an observer situated 2 m from the wall. If the visual angle between the upper and lower edges, respectively, is 10 degrees, what is the height of the picture?

9. A step 6 m. long rests on a vertical wall so that the foot of the ladder is 1.5 m. the base of the wall. What is the angle that the ladder is against the wall and how high the wall reaches the ladder?

10. The lengths of the shadows of two vertical posts are 22 m. and 12 m. respectively. The first pole is 7.5 m. higher than the second. Find the sun's elevation angle and the length of each post.

11. A tree to 12 m. is high on the side of a stream. The angle of elevation of the tree, from a point at 180 m. is 3 degrees. Determine if the stream is above or below the designated point and calculate the level difference.

12. What is the height of a hill, if its elevation angle, taken from its base, is 46 degrees, and taken from a distance of 81 m. is 31 degrees.?

13. On a reef is a beacon whose height is 7.5 m. From a point on the beach can be seen that the angles of elevation to the top and bottom of the lighthouse is 47 degrees and 45 degrees. Calculate the height of the reef.

14. On a horizontal plane, a mast is attached by two wires, so that the straps are on opposite sides. The angles formed by these ties to the ground are 27 degrees and 48 degrees. If the distance enters the wedges is 50 m. How cable has been spent?, what is the height at which the cables are held?

15. From the top of a tower of 200 m. above sea level, the angles of depression of two boats are 47 degrees and 32 degrees respectively. Determine the distance between these boats.

16. A surveyor at C, find two points A and B on opposite sides of a lake. If C is at 5,000 m. of A and 7,500 m. of B and the angle ACB is 35 degrees. What is the width of the lake?

17. Two rangers discovered the same illegal campfire in the direction N 52 ° W and N 55 º E, from their respective positions. The second keeper was 1.93 km. West of the first. If nearest fire warden is the one to go. Which of them has to go and how far to walk?

18. One area is shaped like an isosceles triangle. The base is off a road and has a length of 562 m. Calculate the length of the sides if they form an angle of 23 degrees.

19. A ship leaves a port and travels to the West. At one point rotates 30 degrees North on the West and travel 42 km. further to a point 63 km far. port. How far is the port to the point where he turned the boat?

20. From the top of a tower 300 m. there is a high plane and elevation angle of 15 degrees and a car on the road, on the same side of the plane, with a depression angle of 30 degrees. At the same moment, the car driver sees the plane at an angle of elevation of 65 degrees. If the plane, car, and the observer lie in a vertical plane: calculate the distance between the aircraft and automobiles, also calculate the height at which the plane flies at that time.

21. A triangular area is demarcated by a stone wall of 134 meters, one in front of 205 m. down the road and about 147 m. What angle does the fence with the road?

22. A ladder, whose foot is on the street, forming an angle of 30 degrees with the ground, when its upper end rests on a building on one side of the street, forming an angle of 40 degrees when it meets in a building across the street. If the length of the ladder is 50 m, what is the width of the street?

23. A tree has been broken by the wind so that the two parties are to land a right triangle. The upper part forms a 35 degree angle with the floor, and the distance measured on the ground, from the trunk to the height is 5 m. fall. find the height that was the tree.



24. An observer detects an unidentified flying object statically located at a point space. The observer, using a rangefinder and a sextant, determines that the UFO is at 4460 m. at an elevation angle of 30 degrees. Suddenly the UFO descended vertically to land on the surface. Determine how far the observation point down this object and how far it should go down to touchdown.

25. The angle of one corner of a field in a triangular shape, measures 73 degrees. If the sides, among which is that angle, has a length of 175 feet and 150 feet, determine the length of the third side.

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Trigonometric Trigonometric ratios


The triangle is the key figure metric description for any surface.
Surveyors in purely practical applications, reduce their problems to triangulations.
The theoretical mathematician, in his analysis, often also used this basic polygon
should consider a number of previous definitions:
• A segment x is the mean proportional between two a, b when checked:
·
· Given a point P ay line is called projection of point P on the line to the point P 'defined as the intersection of the perpendicular drawn from P aa and the same line a.
· Projection of a segment MN is on a straight segment M'N ' whose endpoints are the projections of the ends M, N given segment on the line a.
• In a right triangle, sine of an angle is the ratio (the ratio) between the opposite side and the hypotenuse, the cosine of an angle is the relationship between adjacent and the hypotenuse side, the relationship between the sine and cosine an angle (or equivalently the ratio of opposite and adjacent) is the tangent of that angle.
• Although defined in a right triangle, the values \u200b\u200bof sine, cosine and tangent of an angle are fixed values \u200b\u200bthat are fully tabulated in trigonometric tables.
· The reasons sine (sin), cosine (cos) and tangent (tan) and their inverses: cosecant (cosec = 1/sen), secant (sec = 1/cos) and cotangent (cotg = 1/tg) are called trigonometric ratios.




Be the triangle ABC (rectangle in A)
angle
trigonometric ratio 0 º 30 º 45 º 60 º 90 º
Seno 0 0.5 0.707 0.866 1
Cosine 1 0.866 0.707 0.5 0
Tangent 0 .577 1 1, 73 oo

Theorem: Theorem
leg.
In any right triangle verifies that one leg is a mean proportional between the hypotenuse and the projection of that leg on the hypotenuse:

height theorem.
In any triangle, the height is a mean proportional between the two segments that divides the hypotenuse:

Pythagorean Theorem.
In any right triangle is true that the square of the hypotenuse equals the sum of the squares of the legs:
a2 + b2 = c2
From the definitions of the trigonometric ratios follows:
b = a sin B = a
cos C c = a cos B = a sin C
tg B b = c, c = b tg C

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Sources

Historians agree that it was the Greeks before Socrates initiators of trigonometry. Thales of Miletus, one of the Seven Sages of Greece, is credited with the discovery five geometric theorems and their participation in the determination of the heights of the pyramids of Egypt using the relationship between angles and sides of a triangle. Hipparchus, Greek astronomer and mathematician notable, systematized these concepts in a table of trigonometric chords that are now the basis of modern trigonometry. For his work is considered the father or founder of trigonometry.
is considered to Hipparchus (180-125 BC) as the father of trigonometry mainly for his discovery of some of the relationships between sides and angles of a triangle. Also contributing to the consolidation of trigonometry Claudius Ptolemy and Aristarchus of Samos who applied it in his astronomical studies. In 1600, professor of mathematics at Heidelberg (the oldest university in Germany) Pitiscus Bartholomew (1561-1613), published a text with the title of trigonometry, which develops methods for solving triangles. The French mathematician François Viète (1540-1603) made important contributions to finding multiple angle trigonometric formulas. Trigonometric calculations received a boost thanks to the Scottish mathematician John Napier (1550-1617), who invented logarithms in the early seventeenth century. In the eighteenth century, the mathematician Swiss Leonhard Euler (1707-1783) made a science of trigonometry in addition to astronomy, to turn it into a new branch of mathematics. Originally
, trigonometry is the science whose object is the numerical (algebraic) of the triangles. The six key elements in any triangle are its three sides and three angles. When you know three of these elements, provided that at least one being the one hand, teaches trigonometry to solve the triangle, that is, to find the other three elements. In this state of trigonometry defined the trigonometric functions (sine, cosine, tangent, etc.) Of an acute angle in a triangle rectangle, as the ratios between two sides of the triangle, the domain of definition of these functions is the set of values \u200b\u200bthat can take the angle [0, 180].
HOWEVER, the study of trigonometry does not limit its application to the triangles, geometry, navigation, surveying, astronomy, but also for the mathematical treatment in the study of wave motion, vibration, sound, alternating current, thermodynamic atomic research, etc.. To achieve this, we must broaden the concept of a function trigonometric function of a real variable, rather than simply a function of angles.

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trigonometry trigonometry, What is?

The basic problem of trigonometry is something like this:

is near a wide river and need to know the distance to the other side, say to the tree marked in the drawing by the letter C (for simplicity, ignore the 3 rd dimension). How to do it without crossing the river?

The usual way is as follows. Drive two stakes into the ground at points A and B and with a tape measure the distance c between them (the "base").

An old surveyor's telescope (theodolite).
Then remove the post from point A and replace it with a surveyor's telescope as the one shown here ("theodolite"), with a plate divided into 360 degrees, check the direction ("azimuth") that points the telescope. Pointing the telescope to the tree first and then to the pole B, measures the angle A of triangle ABC equal to the difference between the numbers that you have read the azimuth plate. Replace the post, move the theodolite to point B and measured in the same way the angle B.
length c of the base and the two angles A and B are all you need to know the triangle ABC, enough, for example, to build a triangle of the same shape and same size, in a more convenient place. Trigonometry (trigon = triangle) at first was the art of calculating the lost information by simple calculation. Given sufficient information to define a triangle, trigonometry is used to calculate the other dimensions and angles.

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What is trigonometry?

Trigonometry is a branch of mathematics of ancient origin, whose etymological meaning is "the measurement of triangles." It derives from the Greek word τριγωνο "triangle" + μετρον "measure"

in principle Trigonometry is the branch of mathematics that studies the relationships between angles and sides of triangles. For this it uses the trigonometric ratios, which are frequently used in engineering calculations. Overall, trigonometry is the study of the sine, cosine, tangent, cotangent, secant and cosecant. Directly or indirectly involved in the other branches of mathematics and applies in all areas where measures are required accuracy. Trigonometry applies to other branches of geometry, as is the case study areas in the geometry of space.

has many applications: triangulation techniques, for example, are used in astronomy to measure distances to nearby stars, in measuring distances between geographic points, and satellite navigation systems.