Andrew Sciacchitano, Jocele Shelton, Jacob Montgomery, Jordan McClure

Trigonometry


external image Angling1.jpgexternal image trigonometry.gifexternal image trig_intro.gif





THE HISTORY OF TRIGONOMETRY:

Hipparchus of Rhodes
The first trigonometry problem may have been solved by Hipparchus of Rhodes (ca. 190-120). His problem dealt with why the seasons are different lengths. Hipparchus used the observed lengths of each of the seasons to determine the arc length traveled by the sun in its orbit during each season. He then found the lengths of the chords that connect the sun’s position at the ends of the seasons. These lengths enabled him to determine how far the earth was from the center of the sun’s orbit. Hipparchus also was able to create a table to approximate the chord lengths of a circle.

Finding Chord Lengths
Euclid may have been the first person to show how to calculate other chord lengths when he determined the lengths of the sides of regular inscribed pentagons and decagons. Using the foundation that Euclid started, Ptolemy of Alexandria (90–168 CE) constructed a table of chord lengths for a circle of radius 60 in half-degree increments. Ptolemy’s table is equivalent to a table of sines in quarter-degree increments. His calculations were carried out to seven-digit accuracy.
Indian astronomers in the third fourth or fifth century CE started the shift to the half-cord or sine. The use of sine was then adopted by Arabic mathematicians, later translated into the Latin word Sinus, literally meaning half-cord. From the Latin root is where we get the English Sine.



chord of a circle.jpg


The Unit Circle
The development of the unit circle can be credited to Leonhard Euler (1707-83). It was Euler who decided that for the purposes of calculus the radius should be fixed at 1. He realized that measuring the arc and line lengths in the same units were and that if the radius is one then the circumference is 2 Pi. It seemed that he was using radians, but radians wouldn't start to be used for almost a hundred years after Euler's death.


unit.jpg



Triangle Trigonometry
Triangle trigonometry started with the simple problem of finding the length of a shadow cast by an object at certain angles of the sun. Finding the answer to this question were mathematicians such as Ptolemy and Al-Khwarizmi of Baghdad (ca. 790–840) whose name is the root for the word algorithm and whose book, al-jabr, is the root for the word algebra. The early triangle trigonometry problems led to the formation of trigonometric identities. Because the length that we're trying to find in this problem is tangent to the circle, this function is now known as the tangent. The secant, arising from the Latin secantem, meaning “cutting,” is the length of the radial line segment cut off by the tangent. The cosine, cotangent, and cosecant are the corresponding line segments for the [[#|complementary angle]].

Using trigonometry to solve for right triangles became prominent with the posthumous publishing of Johann Müller's De Triangulis Omnimodis (On Triangles of Every Kind). Müller, Georg Rheticus, and Bartholomew Pitiscus also used trigonometry and similar triangles to solve for an unknown side of any given right triangle for which one of the acute angles and one other side are given.

History of trig

[[#|Applications]] of Trigonometry:

Architecture
In architecture, trigonometry plays a massive role in the compilation of building plans. For example, architects would have to calculate exact angles of intersection for the components of their structure to ensure stability and safety. Some instances of trigonometric use in architecture include: arches, domes, support beams, and suspension bridges.Architects use trigonometry to calculate structural load, roof slopes, ground surfaces and many other aspects, including sun shading and light angles.Architecture remains one of the most important sectors of our society as they plan the design of buildings trigonometry is highly used to insure its longevity,

Astronomy
Astronomy has been studied for millennia by civilizations in all regions of the world. In out modern age being able to [[#|apply]] Trigonometry to astronomy helps us calculate distances between stars and learn more about the universe. Menelaus' Theorem helps astronomers gather information by providing a back drop in spherical triangle calculation.

Geology
Trigonometry is used in geology to estimate the true dip of bedding angles. Calculating the true dip allows geologist to determine slope stability.

Navigation
Trigonometry is used in navigation to find distances of the shore from a point in the sea. It is also used in oceanography in calculating the height of the tides in the oceans. We can one day use it in space travel because we use trig in finding the distance between celestial bodies in space. We can also find people and places using triangularization the process of determining the location of a point by measuring angles to it from known points at either end of a fixed baseline.

Physics
Trigonometry plays a huge role in all varieties of physics. For example, consider that radio, microwave, and electromagnetic waves are all measured and graphed with trigonometric functions such as sine and cosine. Physics concerns itself greatly with three-dimensional space. Mathematics is critical to being able to determine values and help identify the nature of something in the space it exists in. Trigonometry is used so much in physics that it is as prevalent as simple addition and subtraction for the average person.



BASIC IDEAS:

The Unit Circle

Trigonometric functions are defined using the unit circle. The Unit Circle is centered at (0,0) and has a radius of 1 unit. Let Θ (theta) be an angle measured counterclockwise from the x-axis along an arc of the circle. Then cos Θ is the horizontal coorinate of the arc endpoint, and sin Θis the vertical coordinate, as shown below.

trig_functions_in_the_unit_circle.gif

Degrees

A circle is divided into 360 parts called degrees. There are many theories about the use of degrees as a measure of the angles around a circle. Some say that the use of the number 360 corresponds to the amount of days in a year rounded down to the nearest 10. (for more theories click here). While the background of degrees is debatable the fact that they exist and can be used to measure angles in many different forms has provided a great opportunities.
external image degrees-360.gif

Radians

Along with degrees we also can measure the angles around a circle in units called radians. Unlike degrees radians have a more mathematical background. In order to properly describe radians we must first look at the circumference of a circle. The formula for the circumference of a circle is C = 2PiR, where R = radius. If the radius is 1 like in the case of the unit circle above we see that the circumference of the circle will be 2Pi. This in fact is the total distance around the circle. If we divide the circle in half with a horizontal line through the center we will notice that the top angle made by the line is Pi and the bottom angle made by the horizontal line is also Pi , thus we can see that adding these two halves together gives us 2Pi. But, we have not found what 1 radian looks like. As you may have noticed the word radian is related to radius. If we draw a horizontal line from the center of the circle to the right edge of the circle we have what is called the radius. To show 1 radian we need to leave the radius connected to the outside edge of the circle and rotate the left end until it is outside the circle. Next we need to curve that line with the circle in a counter-clock wise direction. This arc length is 1 radian. If we continue to add the same arc length to the circle we will find that the circle is about 6.2831 radians. While this seems like a messy number we can clean it up simply by looking back at our work on the circumference of a circle. We found that the circumference of the unit circle is 2Pi and when we find the value of 2Pi we see that it is in fact 6.2831, the same number we found for the number of radians in a circle.
Below is a graph of all of the radian values corresponding with their matching degree values.
external image 300px-Degree-Radian_Conversion.svg.png

Right Triangles

Right triangles are often used in trigonometry, as seen above. A right triangle has three sides, hypotenuse, adjacent to a given angle Θ and opposite to given angle Θ as shown below.


external image trig-right-triangle-names.gif
The angle can be found using the definitions of the trigonometric functions. There are many mnemonics used to help students remember these, such as the most commonly used "SOHCAHTOA" where sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, and tangent equals opposite over adjacent. Another mnemonic used is "Tommy On A Ship Of His Caught A Herring."The trigonometric functions are cosine (cos x), sine (sin x), tangent (tan x), cosecant (csc x), secant (sec x), and cotangent (cot x). Below are some of the trigonometric functions.

external image trid4.gif

(an example of how the values of sine, cosine, tangent, cosecant, secant, and cotangent change as theta changes can be found on the webpage TouchTrigonometry.org)


-links given from http://mybb.gvsu.edu/webapps/portal/frameset.jsp?tab_tab_group_id=_2_1&url=%2Fwebapps%2Fblackboard%2Fexecute%2Flauncher%3Ftype%3DCourse%26id%3D_320057_1%26url%3D

TRIGONOMETRIC IDENTITES:

A mathematical identity (as defined by the Merriam-Webster's Dictionary - www.m-w.com) is an equation that is satisfied by all values of the symbol. Here, we can think of the word symbol as the exact same thing as a variable. For example, though trivial, x = x is an identity because for every value you can substitute for 'x', the equation will always be true. In trigonometry, identities are very important and help us tremendously. They can allow us to put a function in terms of one trigonometric function into terms of another. The most basic, and likely most common, identity is the Pythagorean identity, which states:
pythagorean_trig_identites.gif
(from: http://www.mathacademy.com/pr/prime/articles/trig_ident/index.asp)

While these three equations appear different, they are equal to one another. Divide the first equation by cos^2 and you get the second equation. Divide the first equation by sin^2 and you get the third equation. Other useful identities can be found at TrigIDs.
These identities are put into groups relevant to what they represent. For example, the half angle identities represent what happens when the variable of a trigonometric funciton is divided by two.
The most intriguing and useful of these identities is probably the cos(2x) identity (seen in the 'double angle identities' group in the list of identities). What makes this so interesting is that it takes a function of degree one - (cos(2x)) - and converts it into a function of degree two. The same can be thought of in reverse as well.
Trigonometric identities are very useful when changing the representation of a trigonometric function. For example, if you have the square root of (1 - cosx)/2, then you can simply write this as sin(x/2). Applications of this and similar processes can be seen using all of the other trigonometric identities.

Putting together the things that we now know about trigonometric functions and the Unit Circle, we can construct a very well-defined graphical image of the Unit Circle itself, shown below, which is considered by many to be the base from which all of trigonometry stems.
UnitCircle.gifThe Unit Circle with points labeled in degrees, radians, and cartesian coordinates.


THE LAW OF SINES AND THE LAW OF COSINES:
THE FUNCTIONS APPLIED TO NON-RIGHT TRIANGLES

The Law of Sines is an important link between right triangles and other triangles in Trigonometry. It asserts that a certain ratio holds true in any triangle, namely, that the ratio of the sine of an angle to the side opposite that angle is a constant. The proof is as follows.

Consider the triangle ABC.


Picture_1.png
Now we will drop a perpendicular from one of the vertices of the triangle and label the length of the perpendicular h. Let's consider the angle A. Now, sinA is defined as the side opposite angle A divided by the hypotenuse of ACD, or h/b. So using algebra, bsinA = h. Now also consider angle B. similarly, sinB = h/a. So then asinB = h. Now we have two equal expressions.


asinB = bsinA
(sinA)/a = (sinB)/b

since this will work with any arbitrary perpendicular, we can say with certainty that the ratio will hold true for all sides and all angles of our triangle, therefore


(sinA)/a = (sinB)/b = (sinC)/c

The Law of Cosines is a little bit more complex of a claim for triangles. It asserts that



Picture_2.png
Consider the triangle ABC again, redrawn below.

Picture_3.png
Looking at the triangle ACD and using the Pythagorean Theorem, we can see that

(x^2) + (h^2) = (b^2)
and we know from our Trigonometric Identities that

cosA = x/bbcosA = x
We will next examine triangle BCD and using the Pythagorean Theorem, we can see that

((c - x)^2)+(h^2) = (a^2)(c^2) - 2cx + (x^2) + (h^2) = (a^2)(c^2) - 2c(bcosA) + (b^2) = (a^2)(b^2) + (c^2) - 2bccosA = (a^2)
The last line of the equation we have derived is known as the Law of Cosines, and since this property will hold true for any perpendicular, we can conclude that

(a^2) = (b^2) + (c^2) - 2bccosA(b^2) = (a^2) + (c^2) - 2accosB(c^2) = (a^2) + (b^2) - 2abcosC
Making Connections with a Model:Great for Visual Learners
trigidentities.png
There are many connections that can be made with this picture. To get your students started you may show them that sin^2(x) + cos^2(x) = 1. You can see in this picture that all these can be connected and help the students start to formulate their own connections.


Example Problem:
Open: http://www.geogebratube.org/student/m1128

Using the slider, change the size of the angle ‘alpha’. Observe what happens. Answer the questions and try to work out the relationship between the sides, angle and sin, cos, tan.

Set the angle to 20o
a) What is the length of the opposite side? .............................
b) What is sin α =.............................
c) What is the length of the adjacent side? .............................
d) What is cos α =.............................

What connection do you notice between the opposite side and sin α? Can you test this?


What connection do you notice between the adjacent side and cos α? Can you test this?


If the angle was 45 o how could you predict the lengths of the opposite and adjacent sides?




Can you make a similar connection for tan α? (Check the hint box if you are struggling)




Now open: http://www.geogebratube.org/student/m1129
Do your conjectures still work?
What is the problem? Is there anything you can do to make the answers work?




If the hypotenuse was 3, what would you have to do?


Check whether this works here: http://www.geogebratube.org/student/m1130
Can you write formula for each of the following?

sin α =

cos α =

tan α =


pictures from lawofsinespicture, lawofcosinespicture

Educational Trigonometry Links:

Trig In Nature

Trig At Work

Touch Trigonometry




COMMON CORE STANDARDS FOR TRIGONOMETRY:**

Michigan High School Content Expectations

P6 Trigonometric Functions
  • P6.1 :Define (using the unit circle), graph, and use all trigonometric functions of any angle. Convert between radian and degree measure. Calculate arc lengths in given circles.
  • P6.2 Graph transformations of the sine and cosine functions (involving changes in amplitude, period, midline, and phase) and explain the relationship between constants in the formula and transformed graph.
  • P6.3 Know basic properties of the inverse trigonometric functions sin-1 x, cos-1 x, tan-1 x, including their domains and ranges. Recognize their graphs.
  • P6.4 Know the basic trigonometric identities for sine, cosine, and tangent (e.g., the Pythagorean identities, sum and difference formulas, co-functions relationships, double-angle and half-angle formulas).
  • P6.5 Solve trigonometric equations using basic identities and inverse trigonometric functions.
  • P6.6 Prove trigonometric identities and derive some of the basic ones (e.g., double-angle formula from sum and difference formulas, half-angle formula from double-angle formula, etc.).
  • P6.7 Find a sinusoidal function to model a given data set or situation and explain how the parameters of the model relate to the data set or situation.

National Council of Teachers of Mathematics

Expectations: In grades 9–12 all students should
  • Explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them;
  • Use trigonometric relationships to determine lengths and angle measures.
State Standards (As set by the CCSS institution):
Define trigonometric ratios and solve problems involving right triangles
  • Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
  • Explain and use the relationship between the sine and cosine of complementary angles.
  • Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Apply trigonometry to general triangles
  • Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
  • Prove the Laws of Sines and Cosines and use them to solve problems.
  • Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

Note: No reference needed because of non-commercial use which is allowable under the CCSS institution terms of use