**USES OF TRIGONOMETRY**

TRIGONOMETRY

How steep is this hillside? How high is that mountain? These sorts of questions pop up all over in geosciences - from plate tectonics to maps to ocean waves, and they require you to find either an angle or a distance. To do this, we often use trigonometry.

Trigonometry - is a branch of mathematics dealing with the relations of the sides and angles of triangles and with the relevant functions of any angles.

In this presentation we will tackle the use of Trigometry in three things, namely, ASTRONOMY, TARGET SHOOTING, and CARTOGRAPHY.

Target Shooting as a Sport

Shooting has evolved from the spear and the projectile throwing contest to the present Olympic game. From using it just for necessity in the pre-historic era, it gradually became a sport with 20 million participants in the United States alone.

Before, shooting was all spears and sticks but when the 10th century came, it became a social and recreational sport. In the 13th and 14th centuries, shooting clubs were formed by Germans and only men can be a member. It was bows and wheel lock muskets that were used but by the 16th century, firearms with rifled barrers were used in matches held publicly.

Connecting this to Trigonometry, the principle of right triangles were used in order to get the right amount of angle, measuring the right distances so that the shooter may shoot the bull's eye so to speak.

Cartography

Cartography (from Greek word, khartes = papyrus (paper) and graphein = to write) is the study and practice of makingmaps. Combining science, aesthetics, and technique, cartography builds on the premise that reality can be modeled in ways that communicate spatial information effectively.

Math helps cartographers with map scale, coordinate systems, and map projection. Map scale is the relationship between distances on a map and the corresponding distances on the earth's surface expressed as a fraction or a ratio, coordinate systems are numeric methods of representing locations on the earth's surface, and map projection is a function or transformation which relates coordinates of points on a curved surface to coordinates of points on a plane.

Maps may include information about elevation, latitude, longitude, distance, population density, climate data, and other relevant information, which of course uses mathematical approaches by using Trigonometry, Algebra, Calculus, and the like. The information emphasized on a map may vary depending on its expected use.

Sample Problem:

A, B and C are three ships. The bearing of A from B is 045º. The bearing of C from A is 135º. If AB= 8km and AC= 6km, what is the bearing of B from C?

tanC = 8/6,

C = arctan 8/6

so C = 53.13º

y = 180º - 135º = 45º (interior angles)

x = 360º - 53.13º - 45º (angles round a point)

= 262º (to the nearest whole number) -

Sample Problem

Assume we have zeroed a rifle at 300 yd on a level range and we are shooting at a target on a slant range of 300 yards.

Assume the slope angle

THETA is 30 degrees

. The

cosine of 30 degrees is 0.87

. The

horizontal range for the bullet is only 261 yards (300 * 0.87)

. In order to hit the target we should hold the gun as if the target were only 261 yards away not 300 yards. If we shoot where the scope crosshairs intersect the target we will shoot over the target.

SOURCES

http://www.loadammo.com/Topics/April04.htm

http://www.mathsrevision.net/gcse-maths-revision/trigonometry/bearings#sthash.WDp0RbOF.dpuf

http://trigonometryinastronomy.wikispaces.com/Applications+of+Trigonometry+in+Astronomy

http://earthguide.ucsd.edu/virtualmuseum/ita/06_3.shtml

http://en.wikipedia.org/wiki/Cartography

http://www.usashooting.org/about

http://www.ucolick.org/~raja/ay2/f04/hw2ans.pdf

SAMPLE PROBLEM

An astronomer observes a bright star (Altair) that has a parallax angle of p = 0.20 arcseconds. The ﬂux f from Altair is approximately 9.4 × 10−12 times the ﬂux

from the Sun. The distance d from the Earth to the Sun is (1/206265) pc.

What is the distance d to Altair star in units of parsecs (pc)?

Use the formula: d = 1/p, with distance d in units of pc and parallax p in

units of arcseconds.]

Solution:

d = 1/p = 1/0.20 = 5 pc,

ASTRONOMY

Astronomy is known as the study of the moon, stars and other celestial bodies. For NASA, or National Aeronautics and Space Administration in the United States, it is simply the study of stars, planets and space, but there is a deeper meaning to it if we will apply Trigonometry. If we will apply Trigonometry in astronomy, we can discover the distances between two celestial bodies and distances of spherical entities which is in line with spherical trigonometry.

In simple trigonometry, in calculating how distant he earth orbits the sun in a radius in which the earth is opposing its orbits and appears as a star acting in a dissimilar position among the other stars. The angular extent of an elliptical arc among the setting of space that the star appears to track is called the parallax, symbolized by the Greek letter Θ. In astronomy, the formula tan Θ=r/d is used where the earth’s orbit is the radius or r and is equal to 1 A.U., the distance of the start pertains to d, and the tangent of the triangle is tan. In span of months, astronomers can compute for the distance, as well as determining the parallax by evaluating pictures of heavenly bodies.

ESTONILO, Ma. Katrina Gale Z.

DELOS REYES, Tyra P.

ESPIRITU, Nielzen Jude Q.

JIMENEZ, Angelica Faith C.

Ferriols, Christopher Lawrence C.

MATH112 - NO3