LOGARITHMS AND EXPONENTIAL

NAOMI OSAIGHALE

NOVEMBER 5TH 2017

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Contents

1 Origins and Development of Trigonometry 31.1 Europe And Trigonometry . . . . . . . . . . . . . . . . . . . . . . 31.2 Modern Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Connection Between Trigonometry and Trigonometric Func-tions as Circular Functions 4

3 Trigonometry 43.1 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . 43.2 The fundamental Relationship of sin2(x) + cos2(x) = 1 . . . . . 5

4 ANGLES 54.1 Radian Measure and Degree Measure of Angles . . . . . . . . . . 5

5 Graphs of All The Trigonometric Functions 75.1 The Phase connection between Cosine and Sine . . . . . . . . . . 12

6 Addition Formulas for trigonometric 126.1 The Phase connection between Cosine and Sine . . . . . . . . . . 126.2 The Limit of sinx/x as it Approaches 0 . . . . . . . . . . . . . . 13

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1 Origins and Development of Trigonometry

The term ”Trigonometry” comes from the Greek word of ”trigonometria” whichmeans triangle measuring.The modern word sin comes form the Latin wordSinus which means bay or fold. Egyptians, Babylonians, Indians and Chinesehad an idea of trigonometry. The construction of the pyramid showed theEgyptian’s knowledge of trigonometry. Hipparchus (c. 190–120 BCE) was thefirst to construct a table of values for a trigonometry functions. He consideredevery triangle planar or spherical as being inscribed in a circle, so that eachside becomes a chord (that is, a straight line that connects two points on acurve or surface). To compute the various parts of the triangle, one has to findthe length of each chord as a function of the central angle that subtends it-or,equivalently, the length of a chord as a function of the corresponding arc width.But Hipparchus was more interested in the three stars o n the celestial sphere.

1.1 Europe And Trigonometry

In the middle ages Europe had no idea of trigonometry but the Arabs andJewish scholars living in Spain introduced it to them. The first table of tangentand cotangent was done by abash Al-asib aka “the Calculator”, who wroteon astronomy and astronomical instruments. Another Arab astronomer, Al-Battani (c. 858–929), gave a rule for finding the elevation of the Sun above thehorizon in terms of the length s of the shadow cast by a vertical gnomon of heighth.Al-Battani’s rule, s = h sin (90 )/sin , is equivalent to the formula s = h cot. Based on this rule he constructed a “table of shadows”—essentially a table ofcotangents—for each degree from 1 to 90. It was through al-Battani’s work thatthe Hindu half-chord function—equivalent to the modern sine—became knownin Europe.

1.2 Modern Trigonometry

Issac Newton (1642–1727) discovered the power series of sine and cosines.

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In 1722 Abraham de Moivre (1667–1754) derived, in implicit form, the fa-mous formula (cos ø + i sin ø) n = cos nø + i sin nø,which allows one to findthe nth root of any complex number.

2 Connection Between Trigonometry and Trigono-metric Functions as Circular Functions

Circular functions are part of the set of trigonometric functions. Whereastrigonometric functions consist of domains that are sets of angles and rangesthat are real numbers, circular functions have domains that are sets of numbersthat correspond to the angles of the trigonometric functions (in radians). Circu-lar functions are commonly used in geological and structural engineering studiesto create mathematical models of rock folds. This set of functions is considered”circular” because the functions describe a point on a circle as the function ofan angle, which is defined by radians. Radians measure the arc length of a givencircle.

3 Trigonometry

Trigonometry is basically the calculation part of geometry. This is where youapply your knowledge of triangles from geometry and use the resulting formulasto help you solve problems.Trigonometry is only done on right angle trianglesi.e. it has a 90o angle. A triangle has three sides to it . a which is the adjacent,b the opposite ans c the hypotenuse . These three sides make trigonometry.Sine,cosine, and tangent are all different ratios of these three sides.

3.1 Trigonometric Functions

There are three main trigonometric functions, Sine in relation to a triangle isthe ratio of the length of the side that is opposite that angle to the length of thelongest side of the triangle while in relation to the unit circle it is the extendedreal value in terms of the length of a certain line segment in a unit circle. Thesecond main trigonometric function is Cosine of an angle is the ratio of thelength of the adjacent side to the length of the hypotenuse. Cosine is comple-mentary to Sine. And the last Trigonometric function is Tangent is the ratio ofthe length of the opposite side to the length of the adjacent side in relation toa triangle and in relation to a Circle it is a line segment tangent to the circle,that is the line that touches the circle.

The trigonometric functions are used in right angles triangles. To find theSine of a Triangle then it will be Opposite/HypotenuseCosine : Adjacent/HypotenuseTangent : Opposite/AdjacentThe acronym SOHCAHTOA was made for memorization. Other Trigonometric

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Functions are derived from these three. Co secant is the reciprocal of Sine,Secant is the reciprocal of Cosine and Cotangent is the reciprocal of Tangent.

3.2 The fundamental Relationship of sin2(x) + cos2(x) = 1

The Pythagoras theorem can use to prove that the fundamental relationshipsin2(x) + cos2(x) = 1 for all x, and why it is true.

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The Unit circle can also be used to prove this. A unit circle is a circle with theradius of 1. The unit circle has a formula x2+y2 = 1 for all random points in theunit circle have coordinates that prove it to be true. Then substituting x withCosine and y with Sine thereby making the equation into sin2(x)+cos2(x) = 1.And derivative of f(x) = Sin2(x) + Cos2(x)f ′(x) = 2Sin(x)Cos(x) + 2Cos(x)(−Sin(x))= 2Sin(x)cos(x) − 2cos(x)sin(x) = 0Since the derivative is zero everywhere the function must be constant.f(0) = Sin2(0) + COS2(0) = 0 + 1 = 1So sin2(x) + cos2(x) = 1 everywhere

4 ANGLES

Angles are formed by two rays that begin at the same point or share the sameendpoint

4.1 Radian Measure and Degree Measure of Angles

A radian can be defined as a unit of angle, equal to an angle at the center ofa circle whose arc is equal in length to the radius.it is a unit of measure of anangle it can be used in place of degrees. A circle is 360 degree when convertedto radians it is over 6 radians. One radian is 57.295 degrees.

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The diagram above shows the unit circle in radian and degrees.

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5 Graphs of All The Trigonometric Functions

Figure 5.1 Showing the graph of sines

Figure 5.2 Showing the graph of cosines

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Figure 5.3 showing the graph of tangents

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Figure 5.4 showing graph of co-secants

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Figure 5.5 showing the graph of secants

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Figure 5.6 showing the graph of cotangent

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5.1 The Phase connection between Cosine and Sine

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The connection between Sine and Cosine is that their graphs are the samething, but only shifted 90 degrees or pi/2.

6 Addition Formulas for trigonometric

6.1 The Phase connection between Cosine and Sine

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6.2 The Limit of sinx/x as it Approaches 0

The limit if sinx/x is 1 when the x values are approaching 0. This limit saysthat as x gets small the ratio of sin(x) to x approaches 1. If the ratio of twonumbers is close to 1, then the two numbers are about equal. For small valuefor x, sinx is almost equal to x. The limit is 1 because its in radians, if it wasnot in radians it would have not been 1.

Figure 6.2 showing the graph of sinx/x

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