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Unit circle and the basic trigonometric identities

Unit circle : Different approaches and researches on the topic.

#title#Unit circle and the basic trigonometric identities#title#


In various textbooks, the unit circle is approached in a way that introduces students to trigonometric functions. In most coursework definition, functions are defined according to the coordinate points where an angle subtends a circle. Aside from the unit circle, basic trigonometry needs to be introduced by defining the trigonometric functions which connect to the definition of ratios and adoption of ratio methods. The learning of students is not a haphazard affair but it is constantly managed by the methods chosen to convey the learning concepts and ensure that the necessary skills are developed with a number of learners and teachers influence. Since the advent of mathematics, two methods have been used to introduce trigonometry in schools. The methods are :

  • Ratio method: This is a method whereby trigonometric functions get defined as ratios of pairs of sides of a right angled triangle. For example, the angle sine is defined by the ratio of the opposite side length to the hypotenuse length. Students are taught to always remember the ration definition by a mnemonic SOHCAHTOA, which the first part, SOH stands for Sine=opposite + hypotenuse. 
  • Unit circle method: This unit circle method is defined by the sine and the cosine which are the point’s x and y coordinates. Although this definition applies to all circle quadrants, recent textbooks only use the first quadrant. To determine the angles and lengths in right-angled triangles, a triangle is compared with standard reference triangles and the properties of the same type of triangles are used in the calculation. 

In mathematics, trigonometric functions of all sorts are covered in high-level courses like in Differential equations. To understand the unit circle, it is important to convey to learners the functional nature of a trigonometric function. There are a variety of trigonometric identities. They include ratio identities, cofunction identities, Pythagorean identities, reciprocal identities and symmetric identities. Learners should master various basics like the trigonometric ratios which they also have to indicate in their research proposal during their course. They include sinθ=opposite/hypotenuse, secθ=1/cosθ, cotθ=1/tanθ, tanθ=opposite/adjacent, cosθ=adjacent/hypotenuse and cosecθ=1/sinθ. The basics of converting radians to degrees are also crucial. They include 2πradian=360 degree, radian= (180/π)degree, 360 degree=2πradian, radian=(360/2π)degree, degree=(2π/360)radian and degree=(π/180)radian. In a full circle, there are 2π radians. The special angles are π/6, π/3, π/4, π/2, π, and the multiples of all like 5π/6. The most commonly used trigonometric identity is the Pythagorean identity which is sin2θ + cos2θ = 1. This identity is very important since it sets an expression that involves trigonometric functions that are equal to one. Being simplified in an equation makes it helpful in solving equations.

Unit circle : The History and the Modern Situation of the problem.

Unit circle and trigonometric identities are a branch of geometry. This branch is quite different from the synthetic geometry of the old Greeks and Euclid because of its computational nature. In the measurement of two angles and the length between them, the problem to be solved is the calculation of the remaining angle. Every trigonometric calculation involves measuring angles and making some trigonometric computations. The current trigonometric functions are cosine, sine, tangent and their reciprocals but in the old Greek trigonometry, functions that were used include the chord and some other intuitive functions. The term trigonometry is derived from a Greek word “trigonometria” which means measuring of triangles. The study of “trigonometeria” started in Hellenistic mathematics and reached India as part of Hellenistic astronomy. Measuring of angles was done for the first time by the Babylonians but trigonometry started in Greece.

The first application of unit circle and trigonometry was in astronomy. This application of the unit circle combination has been explained severally by students in their 1000 words essay assignments. In this application, angles calculations in the celestial sphere needed a particular kind of trigonometry and geometry that is used in the plane. The sphere trigonometry was known as “spherics” and it comprised one part of quadrivium of learning. Currently, the subject is known as “elliptic geometry”. Trigonometry has solved the problems encountered in spherics rather than those experienced in plane geometry. The other original application of trigonometry was in angle measurement by the Babylonians. This took place sometime before 300 B.C.E when degrees were used to measure angles. Babylon numerals were based on a number 60 and so they took that unit measure to be 60 degrees. They then divided it and perhaps those 60 degrees were taken as a unit because a chord of 60 degrees is equivalent to the radius of a circle. Hipparchus of Nicaea is one of the first people to apply trigonometry. He was an astronomer and though the Greeks, Egyptians and Babylonians knew a lot of astronomy before him, he is regarded as the first main person to apply trigonometry and the first chord table is attributed to him. It is always hypothesized that Archimedes and Apollonius came up with tables of chord before Hipparchus, but are no references to such earlier tables. The twelve books of tables of chords were written by Hipparchus in about 140 BC and that is why he is also regarded as the founder of trigonometry. Another Greek mathematician by the name Menelaus was the next one to produce the table of chords in about 100 AD. Menelaus used to work in Rome and had produced six books of tables of chords which have however been lost but his spherics work still exists. He proved the property of plane triangles. Still relating to the origin of unit circle, Claudius Ptolemy is another astronomer who did mathematical collection. This was an astronomy work that included a mathematical theory that was applicable in astronomy. It included a trigonometric table, ½ degrees to 180 degrees angles table of chords in increments of ½ degrees. Those chords were rounded up to two sexagesimal places in an accuracy of about five digits. He applied the geometry necessary to come up with the tables and calculated the chord of 72 degrees together with the chord of 60 degrees. He took the radius to be 60 which gave crd 12 degrees, then crd 6 degrees, crd 3 degrees and crd ¾ degrees. Finally, he applied interpolation to get crd 1 degree and crd ½ degrees.

The unit circle and trigonometry learning relies on geometry as seen in every writing an opinion essay on this topic. The cosine laws for example, originated from synthetic geometry preposition. The prepositions are named II.12 and II.13 of the Elements. Currently, the trigonometry problems need new synthetic geometry developments. Ptolemy’s theorem is a good example that provides rules for chords of the difference and sum of angles and that corresponds to the difference and sum formulas of cosines and sines. The cosine came about during the development of modern trigonometry by Muslim mathematicians in the middle ages. Al-Battan a Persian astronomer generalized the results by Euclid to spherical geometry in the beginning of the 10th century which allowed him to compute the angular distance between stars. During the 15th century, Al-Kashi in Samarq calculated the trigonometric tables to high accuracy and came up with the first explicit statement of the law of cosines in a format that was suitable for triangulation. Up to date, the cosine law is still known as the Al-Kashi theorem in France. This theorem was popularized in the western world in the 16th century by François Viète. In the 19th century, the modern algebraic notation permitted the law of cosines to be written in the symbolic form which is still used today. Currently, there are six trigonometric functions which are thought of as length related to the circle. Cosine and sine are the distances on a circle from an axis to the terminal ray of angle theta. Tangent and cotangent functions are line segment lengths tangent to the circle from the axis of angle theta terminal ray. And finally, the secant and cosecant are the rays lengths or simply secant lines that run from the circle origin to its intersection with tangent lines

Unit circle : Who invented it and where it is used.

No one in particular took credit for having discovered the unit circle, but its features and aspects were discovered by various people. The unit circle does not have any known creator but was rather an amalgamation of many ideas from early mathematicians and astronomers. The contributors to the technology of unit circle include the Greeks, Egyptians and Babylonians. In 1637, Rene Descartes discovered that the unit circle lies on a Cartesian coordinate system. This is a method of finding points in a plane by using two points that are known as the y coordinate and the x coordinate. The Cartesian coordinate system is found in the Euclidean plane which was also discovered by Euclid who was a mathematician from Greek. Measurements of angles were represented by Radians. Radian concept was created by an English Mathematician by the name Roger Cotes in 1714.

A unit circle is a circle with a center at the origin and its radius is one. Its circumference is 2/7 and its arc is the same length as that of a central angle measure that intercepts the arc. The unit circle is the source of generating trigonometric function graphs. This comes about when the unit circle ’s quadrants are placed in a numerical order and horizontally. This graph creation process is known as “unwrapping” the circle. If a point on a circle is at the terminal side of an angle that is a standard position, the sine of that angle is automatically the y coordinate of that particular point while its cosine is the x coordinate of the same point. This type of a relationship has many practical uses that concern the arc length and the circle. If you have an arc with one endpoint at (1, 0) which extends in a counterclockwise direction, the other end point will be determined if the length of the arc is known. When you are given the length of an arc, the other end point will be calculated by (cos(s), sin(s)) coordinates.

Many at times, the unit circle is drawn according to x 2 + y 2= 1 equation. But the unit circle can also be drawn according to x = cos(s), y = sin(s) equations whereby x represents the arc length that starts at (1,0). When working with angles in all the four quadrants, the trigonometric ratios for those angles are calculated in terms of x,y, and r values where r is the circle radius which corresponds to the hypotenuse of the right triangle for the angle. Every two right triangles with an identical base angle θ (“theta”) are always similar in the technical sense where they have their sides in proportion. This similarity is seen more when the triangles are nested. Similarity or proportionality simply implies that the trigonometric ratios from the nested triangles will be the same. Trigonometric ratios of a similar angle θ are the same even when the particular numbers of the two triangles’ sides are not the same. This is to emphasize that in trigonometric ratios, the angle θ is what matters and not the particular triangle where the angle was obtained from.

The unit circle makes other mathematical parts neater and easier. The unit circle is very useful particularly when you have; for any angle θ, the trigonometric values for the cosine and sine are cos(θ) = x and sin(θ) = y. With this and the fact that the tangent is defined by tan(θ) = y/, you can calculate x and y and quickly prove that tan(θ) is also equal to the ratio sin(θ)/cos(θ). Trigonometric ratios are used to calculate angles and lengths of right angled triangles. Most of these uses are outlined in trigonometric functions dissertation conclusionevery time. They are used in navigation, physics and engineering. One of its common uses in elementary physics is in resolving a vector Casey Trenkamp. The cosine and sine functions are mostly used to model periodic function phenomena like light waves and sound, sunlight intensity, position and velocity of the harmonic oscillators, temperature variations throughout the year and day length. Simple harmonic motion models a number of natural phenomena like the movement of mass that is attached to a spring and the pendular motion of mass hanging on a string. The cosine and sine functions are uniform circular motions on dimensional projection. Under general conditions, the periodic function f(x) is expressed as the sum of cosine waves or sine waves in the Fourier series.

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