Sine/Cosine Waves Sine, Cosine and Tangent By: Hillary Johnson

&

Ashley Shackford contributed Table of Chords to Trigonometry (an early example of a trigonometry table)Purpose of table of chords: gave method for solving triangles

Some historians go as far as saying he "invented Trigonometry" mathematician, astronomer, astrologer, geographer

born 90AD died 168AD

Wrote 13 different books about the different stars and objects within the solar system

Contributed to idea that earth is round Tangent Tangent comes from Latin word tangens

In present participle is tangere

Means “to touch” or “touching”J.H. Lambert discovered the continued fraction representation of tangent in 1770 J.H. Lambert best known for proving irrationality of pi

First to introduce hyperbolic functions into trigonometry Cosine Sine What are they? Ptolemy Unit Circles - Ratios of the length of the sides in a right triangle Sin =

Opposite/Hypotenuse SOH-CAH-TOA Cos =

Adjacent/Hypotenuse Tan =

Opposite/Adjacent Tan(Angle x)= O/A

ex. Angle= 26

Opposite= 10

Adjacent= x

Tan(26)= 0.487

This is the ratio: 0.487 10

----- = -----

1 x Sin(Angle x)= O/H

ex. Angle= 30

Opposite= x

Hypotenuse= 15

Sin(30)= 0.5

This is the ratio: 0.5 x

--- = ---

1 15 Cos(Angle x)= A/H

ex. Angle= 55

Adjacent= 10

Hypotenuse= x

Cos(55)= 0.573

This is the ratio: 0.573 10

----- = -----

1 x Tan:

0.487 10

----- = -----

1 x

x= 20.53 Sin:

0.5 x

--- = ---

1 15

x= 7.5 Cos:

0.573 10

----- = -----

1 x

x= 17.45 - Each helps find the missing side

lengths within the triangle Sine is half a chord (a chord= line segment in which both endpoints lie on the curve of the circle) A half chord = Rsin(a) where R= Radius and (a)= angle

The meaning of the word sine came from Sanskrit

The word for chord-half was jya-ardha which was shortened to jiva, brought into Arabic as jiba and written in Arabic without vowels as jb

Latin translators selected sinus to translate jb, thinking that the word was Arabic for jaib

In English, sinus translated to sine Sine cont. Sine function commonly used to model periodic phenomena such as: sound and light waves, sunlight intensity and day length

Sine wave: A mathematical curve that describes a smooth repetitive oscillation (oscillation: the repetitive variation of the sine wave) defined as the sine of a complementary angle (90 degree angle - given angle = complementary angle)

naming of cosine was work of Europeans writing in Latin.Co.sinus was suggested by English

Modified by John Newton into cosinus in 1658 A circle with radius of one

In trigonometry: unit circle= circle of radius one, centered at the origin (0,0)

In Cartesian coordinate system: (x,y)= point on the unit circle, x and y= lengths of the legs of right triangle, hypotenuse = 1 Hipparchus Geometry is typically used in Construction, Astronomy, Medical Imaging, Robotics and Computer Graphic Design.

Sine and cosine are used very often in architecture and engineering. Abbreviations “sin”, “cos” and “tan” were

first published in 16th century by French mathematician Albert Girard. occurs in nature; ocean waves, light waves, sound waves

sounds that approximate a pure sine wave: whistling and sound made by a tuning fork By:

Hillary

&

Ashley Thanks for watching!

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# Sine, Cosine and Tangent

A Geometry project on sine, cosine and tangent and all the things about them.

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