ࡱ> ^`YZ[\]  bjbjߍ 3w nnz48FdO ~^4Ddt4LLLLLLL,nS VLqnDD2 \ 0Lx)nn4lNx)x)x) nnKLx)Lx)x)naH5K V x%iI mKNldOIVx)V@5Kx)5KIK$LLx)dOV 4:    A Conceptual Analysis of Trigonometric Functions Ryan Shannon EMAT 6500 Dr. Kevin Moore November 29, 2011 Definitions 1.) Larson, R. (2004). McDougal Littell Algebra 2. a.)Angle and Measure i.) An angle is determined by rotating a ray (half-line) about its endpoint. ii.) The measure of an angle is determined by the amount of rotation from the initial side to the terminal side iii.) A measure of one degree is equivalent to a rotation of 1/360 of a complete revolution about the vertex. iv.) One radian is the measure of a central angle  that intercepts an arc s equal in length to the radius r of the circle. v.) Right triangle trigonometry: Let  be an acute angle of a right triangle. The six trigonometric functions of the angle  are defined as follows: sin- = opp/hyp ,cos- = adj/hyp ,tan- = opp/adj ,csc- = hyp/opp ,sec- = hyp/adj ,cot- = adj/opp The abbreviations opp, adj, and hyp represent the lengths of the three sides of a right triangle. Opp=the length of the side opposite  Adj= the length of the side adjacent to  Hyp=the length of the hypotenuse The functions in the second row above are the reciprocals of the corresponding functions in the first row. vii.) Evaluating trig functions by memorizing cases for special angles 30, 60, 90 and 45,45, 90 viii.) Definitions of trigonometric functions of any angle: Let  be an angle in standard position with (x,y) a point on the terminal side of  sin-= y/r ,cos-= x/r ,tan-= y/x x `"0. csc-= r/y y`"0, ,sec-=r/x ,cot-= x/y Because r cannot be zero, it follows that the sine and cosine functions are defined for any real value of . However, if x =0, the tangent and secant of  are undefined. Similarly if, y=0, the cotangent and cosecant of  are undefined. ix.) Let  be an angle in standard position. Its reference angle is the acute angle  formed by the terminal side of  and the horizontal axis. x.) To find the value of a trigonometric function of any angle , xi. Depending on the quadrant in which  lies, affix the appropriate sign to the function value xii.) To define a trigonometric function of a real number (rather than an angle), let t represent any real number. Then imagine that the real number line is wrapped around a unit circle& .As the real line is wrapped around the unit circle, each real number t will correspond to a central angle  will have a length of t. The point is that if  is measured in radians, then t=. So you can define sin t as ,sin-=,sin-,+2 Similarly for cosine and tangent. xiii.) Domain of the sine and cosine functions is the set of all reals. To determine the range of these two functions, consider the unit circle. Because r=1, it follows that sin t=y and cos t=x. Moreover, because (x,y) is on the unit circle, you know that -1 d" y d" 1. So, the values of sine and cosine also range between -1 and 1. 2.)Axler, S. J. (2009). Precalculus: a Prelude to Calculus. a.) Radians- unit of measure angle such that 2 corresponds to the rotation through an entire circle. b.) Cosine- the first coordinate to the end point of radius of unit circle that makes theta with the positive horizontal axis. c.) Sine the second coordinate to the end point of radius of unit circle that makes theta with the positive vertical axis. d.) Tangent- sine/cosine. Slope of radius unit circle that make angles to theta with positive horizontal axis. 3.) Hass, J. (2011). University Calculus, Early Transcendentals. a.) Radian- in the central angle ACB where A and B are on the length and C is at the origin, with in a circle of radius r is defined as the number of radius units contained in the arc s subtended by the central angle. b.) Given P(x,y) and radius length r, where the angles terminal ray intersects the circle i.)sine: sin = y/r ii.)cosine: cos = x/r iii.)tangent: tan = y/x 4.) Swokowski, E. W., & Cole, J. A. (2008). Precalculus: Functions and Graphs. a.) Radian- is the measure of the central angle of a circle subtended by an arc equal in length to the radius of the circle. b.) Trigonometric functions designated as sine, cosine, tangent, abbreviated sin, cos, tan respectively. Given a right triangle c is opposite right angel, b is the side opposite the angle, a is the side that contains the angle. The symbols sin is used for the ratio b/c cos is used for a/c and tan is used for b/a. Sin = opp/hyp cos= adj/hyp tan = opp/adj 5.) Larson, R. (2007). Algebra and Trigonometry. a.)Radians- measure of an angle in standard position whose terminal side interprets an arc of lengths r. b.) Given a right triangle as listed above. Sin = a/c cos = b/c tan = b/a. 6.) Musser a.) Circumference-let r,d and C be the radius, diameter, and circumference of a circle, respectively. Then d=2r and c= d= 2r b.) Area- the area A of a circle with radius r is A= r2 A Conceptual Analysis of Trigonometric Functions Fore Shadow By reading texts and tutoring students, I have noticed that students face challenges in understanding trigonometric functions. Here I explore the challenges students face and show how students educational backgrounds can leave them with gaps in their understanding of trigonometric functions. Here too, I suggest ways these gaps can be filled and avoided. Gaps in student understanding can be created by the textbook definitions used to teach trigonometric functions; and by disproportionate applications of trigonometric functions to triangles and circles. Also, by using these functions without a strong understanding of radians and the unit circle students will have find issues in understanding. Students need definitions they can apply to understanding and solving trigonometric problems. History of Trigonometry Trigonometry was originally used to show angles in the sky and the arc of circles; later it was discovered that these functions could be used to determine the sides of triangles. Rheticus produced many tables of trig functions, which were not in print until after his death. These tables were done in degree measure (Kupkova, 2005). Trigonometric tables were created over two thousand years ago for computations in astronomy. The stars were thought to be fixed on a crystal sphere of great size, and that model was perfect for practical purposes. Only the planets moved on the sphere. (At the time there were seven recognized planets: Mercury, Venus, Mars, Jupiter, Saturn, the moon, and the sun. Those are the planets that we name our days of the week after. The earth wasn't yet considered to be a planet since it was the center of the universe, and the outer planets weren't discovered then.) The kind of trigonometry needed to understand positions on a sphere is called spherical trigonometry. Spherical trigonometry is rarely taught now since its job has been taken over by linear algebra. Nonetheless, one application of trigonometry is astronomy (Joyce, 1997). It is claimed that the Greek mathematician Hipparchus produced twelve books of the first known tables of chords in about 140 BC. These books sadly didnt survive. Ptolemy, the next author of chords and having Babylonian influence similar to Hipparchus produced an approximation of  to be 3. This was acquired from dividing a circle into 360 degrees and chords into 120 parts. Ptolemy s work gives radius of a circle 60 parts and the perimeter of the circle divided into 360 parts thus, equivalent length units for both radius length and arc measure, and the idea radians. The first appearance of sine as an angle is done by work of the Hindus, Aryabhata in 500, produced sine tables, which were actually thought to be half chords. Bhaskara gave detailed methods for constructing table of sine for any angle in 1150. The 18th century is the work on trigonometric functions done by Euler, Bernoulli, and Cotes (Kupkova, 2005). Definition Analysis Some definitions state that the trigonometric identities are exclusively sides of a triangle, such that sine = Opposite / hypotenuse, cosine = adjacent/ hypotenuse. (Swokowski, 2008) These definitions make no reference to anything other than that which is related to the sides of a triangle. For these definitions to be applicable, the student must be given a triangle, with sides that will yield the other sides. This is a major flaw in the definition where trigonometry is very useful in angle measure and arc length. A break in and understanding would mean that some students would see no application to the functions in the realm of circles, angle measure connected to the arc length of point, and the terminal angle. Fig. 1  By looking at Fig. 1 we are given P(x,y) and radius length r, where the angles terminal ray intersects the circle sine: sin  = y/r cosine: cos  = x/r tangent: tan  = y/x Let  be an acute angle of a right triangle. The six trigonometric functions of the angle  are defined as follows: sin- = opp/hyp ,cos- = adj/hyp ,tan- = opp/adj ,csc- = hyp/opp ,sec- = hyp/adj ,cot- = adj/opp (Larson, 2004) Looking at this definition, we notice by observation that the use of radians and a point on a circle are both included. Also here Larson includes the angle of reference theta. This is a great addition for the student to see where the triangle is formed on the fig.1 by angle theta. That being said, a student would only need to have an angle and a side and would thus be able to produce sides and angles associated with that triangle. This too would create a stronger understanding of how trigonometric functions are a proportion of arc length in radians, and in turn, give the student the ability to apply this concept to both circles and triangles. This definition has the ability to close the gaps that are created from just assigning single uses to the functions. Radians in Real Life Figure 2 below is a unit circle. This particular unit circle allows a student to be able to connect angle measure with radian measure. This is only a start, now it is the responsibility of the teacher show this circles connections to trigonometry. Students typically are more comfortable when dealing with angles in degree measure than radian measure. This discomfort stems from a poor comprehension of what radian measure is and how it is coupled to degrees. This can stem from a fear of fractions, or the length of time a student has spent learning about degrees. This is because students think we live our life in a 360-degree world, which comforts a student to build a reference to what a measure of a degree is. However, we also live life in 2, which is why it is important for students to know radians too. Students struggle with radians also because there is an exact 2 quantity of radians in a circle. However 2 is not rational, the measure of radians is difficult because one radian is roughly 57 times as large as a degree. Thus there are about 6 radians in a circle, the exact value 2pi is 6.28. and is non-terminating and only an approximation can be found (Degrees and Radians). hat a student rotates about the origin of a circle one radian, this rotation will be roughly a 6th of a circle. Breaking a circle into six parts will be harder for a student visualize and harder to recognize. This breaking is harder to understand as we are cultural sound with 360 degrees thus degrees are easier to conceptually understand. The circumference and area of a circle are already defined in terms of radius lengths. Thus, students have already been exposed to radian measure, and this could be a way to approach teaching this topic. By having students see connection to area, and circumference of a circle using trigonometric functions as in arc length, students may close gaps in the concepts of radians. Fig. 2  Radian Definition As shown in appendix c (Larson, 2005) textbook introductions of radians appear before the introduction of trigonometric functions, but each definition of radians is slightly different. For a student to continue to grow in understanding trigonometric functions, an understanding of radian measure is required. As we have noticed in the trigonometric functions a lack of a proper definition will create a gap in understanding. (Axler, 2009) Radians- unit of measure angle such that 2 corresponds to the rotation through an entire circle. This definition leads the reader to assume that radians must contain the unit of pi; as I have tutored, I have noticed more and more that people think radians must contain a pi measure. This definition lacks a statement about the measure being a proportion of the arc length measured in lengths of the radius.  In the circle approach, a circle can be placed on a coordinate system and the center of the circle is the same as the origin. The angle of measure is from x-axis to the terminal side, where the terminal side is the radius of the circle. When looking for a point on the arc of a circle students are informed to find a triangle within a circle to create and opposite, hypotenuse and adjacent. This connection I find faulty, as when students go to deal with sides of the triangle at 180 degrees, they are hard pressed to find three sides, and therefore become lost. Again, this definition seems to encourage the student to focus on these identities as primary used for a triangle inscribed into a circle of radius length. Circles, then radians and arcs, followed by trigonometric functions is the natural progression that a student could follow in learning. These steps bring the student into the concept with thoughts of radians, arc length, and degrees before giving functions that yield new ways to produce these lengths through triangles. This is another way to show popular angles, pi/6, pi/4, pi/6 all the way to 2pi. Trigonometric functions are a relation of the angle that is formed from the terminal angle to that of the arc length that is contained by that angle. Approaching trigonometric functions from a circle standpoint will focus on the radius of the circle. The focus from (Hass, 2011) Given P (x,y) and radius length r, where the angles terminal ray intersects the circle i.) Sine: sin = y/r ii.) Cosine: cos = x/r iii.) Tangent: tan = y/x from this, we can notice that tangent will be y/x or slope. This definition also includes that the sine is that the y component will be divided by the radius informs the student that not only can this be the hypotenuse of a triangle, but actually the radius of a circle, too. This double definition allows a student multidimensional understanding for sine, or cosine as a function of a radius. Hass (2005) addresses that dividing by the radius is a proportion of the point on the circle to the radius of that given circle, where the units will cancel and the proportion will only be between 0 and 1. Article Synthesis In Radians vs. Degrees by Kupkova, this article follows my topic nicely but seems to steer away from radian understanding during learning trigonometric functions. Kupkova states; In our opinion radians should no be introduced in high school. The proper time to introduce them is when the students need them. This statement is very different from my opinion, and as I read this article was a very interested in his beliefs. This source was to provide evidence that students did not need radian measure when being taught trigonometric functions. There is no work on this essay that was done as research. This is merely done by works of another essay and self-ideas. Kupkova in section 2 and 3 of his article states the history and why radians are used. Here he tries to prove that radians are used only in math courses that are far above many students. He is laying down groundwork to mold the reader into thinking that radians should only be taught to the higher more select few. The students that will benefit from radians are only the ones that will further themselves in mathematics. (Section 7) Under this theory we shouldnt be teaching students who are going to be English majors calculus or hardly any mathematics at all.  Radians are an essential part of learning trigonometric functions. Not introducing radians with the use of trig functions would be closing students to all the works of trigonometric functions. Radians are a way to show circles and arc lengths easier. Kupkova conducted a study in section 8 with five questions related to how students and teachers understand radians. The fifth question,  How would you answer the following question of a student:  why should I use  instead of 180? He answers proved that students and future teachers were very unaware of the uses of radians, and why we use them when applying trigonometric functions. This study only strengthens the concept that we need to be teaching both degrees and radians for complete understanding of trigonometric functions. The article by Keith Weber has a more basic approach to trigonometric functions. Defining a procept to be the amalgam of three components: a process that produces mathematical object, and a symbol which is used to represent either process or object(Weber, 2005). This is the understanding that there are two approaches to a situation; Weber gives the example of 82 that is, representing 8 as 8 times 8 or as 64 (2005). Procept is very useful in the world of trigonometric functions when approaching angles. The two types are triangular trigonometry, and unit circle geometry. An educator would hope that a student would construct a right triangle with an angle of 20 and compare the sides, or on a unit circle on the Cartesian plane construct a ray 20 positive of the x-axis and compare intersections. The conducted experiment introduced two groups of students to trigonometric functions. A professor that preferred the more traditional style of teaching taught group 1. That is, his course was lecture-based he would spend most of the time explaining to the students how to do particular exercises and then illustrating how to complete the exercises. The second study of students is taught using Webers design. I break Webers design into the three Rs: Repeat, Reflect, and Reason. The first step procedure, or repeat. In this stage we intro a concept and have students unaware of its real importance repeat its procedure multiple times. In step two reflect, this is the step when we apply application to the concept and have each student understands the process. In step three the students reason the answer and associate the process with a symbol. They are able to reason the result without applying each step (Weber, 2005). His results are not fully proven but are worth noting. The main difference between these two articles is that Weber, addresses both forms of trigonometry. Weber seeks that students understand both forms of trigonometry. He seeks and expects student to apply reasoning and connections of the forms to one another. Kupkova, only places an importance on learning trigonometry in degrees for lower level students, and states that radians should only be left for upper level mathematics. It is important to note that Kupkova doesnt disclose trigonometry with use of radians in a unit circle Standards After looking at how trigonometry should be taught I find it important to see how the high school standards both local and national describe trigonometry education. Here I search the national, and state standards in search for trigonometric functions. What these standards require from the students learning them. How the standards think the topic should be approached and how they differ from each other. How my research would apply to each standard. How textbooks, and my recollection of the topics associate to one another. Georgia Performance Standards In Georgia performance standards the circle is used to define trigonometric function. (MM4A2) Students are introduced to angles and radians. This is showing that the standards are asking that requiring the students know radians and the unit circle before the basic six T functions. The steps that follow are stating that students need to know how trigonometric functions are related to angles, terminal sides, and the unit circle. Section e. MM4A2 specifically states: find values of trigonometric functions using the unit circle. This is showing that the curriculum is aggressively forcing the connection of unit circle and trigonometric functions. I feel that MM4A2 in the GPS is a very obvious step by the state to bring radian and unit circle understanding fore front in learning trigonometric functions. This is interesting that the says nothing about the students need to know anything about how trigonometric functions relate triangles sides with angles. When tutoring the students trouble stems from radians, and trigonometric measure in the unit circle more than the understanding of triangle association such as sides being opposite, adjacent and, hypotenuse an given angle. As students continue in learning trigonometric functions with a lack of basis on radians, and the unit circle they are challenged to find connections in later problems. The GPS does a great job of slowly walking the students though trigonometric functions. MM4A2 is a great example of building math upon itself. Having left out triangle trigonometry is a good idea, however it does shut off a large area of application for trigonometric functions. By leaving out a strict statement about triangle association the standards have understood how students connect trigonometrys use with triangles quickly and efficiently to leave that section exempt. Common Core State Standards The common core state standards have a similar start to the GPS above. The common core F-TF.1. Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. Again, we see that it is stated students need to be familiar with radian measure and unit circle, before being introduced to trigonometric functions. In F-TF.2. Explain how the unit circle in the coordinate place enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angle traversed counterclockwise around the unit circle. This topic states students need to know how the unit circle is related to a coordinate plane Students should understand how radians rotate around a circle and trig functions create a counter clockwise rotation about the unit circle. This section that is left out of the GPS is a very interesting and useful topic. The common core expects teachers to take students further and understand how trigonometric functions rotate about the origin. Through working with students, I have come to realize that it is easier to build upon a definition of a circle than upon a definition of a triangle. Students dont perceive a circle outside of a triangle, but they do see a triangle within a circle. This connection creates a bridge for the students from the world of radians and arc length to the world of triangles and the sides associated with that triangle.  National Council of Teachers of Mathematics NCTM seems to be the most precise version of all standards about trigonometric functions. NCTM requires students to know how to apply trig functions to real life phenomena. Also asking student to apply the functions toe complex numbers polar coordinates and series. NCTM provides examples within its standards to be even clearer to educators what is expected of students. NCTM states that students are bulleted to know the connection between circular and triangular trigonometry. This bullet, I feel is the most important bullet in the curriculum. NCTM is the only one that makes this very clear, and I feel both the GPS and the CCSS need to follow suit. Also the NCTM states what order trig functions need to be taught, and examples of questions that students should answer correctly. These standards place great variety in the hands of the educator while still keeping a clear focus and direction. Textbook Analysis Sorting through texts book seeing how each introduces trigonometry to high school students, there are usually two categories. Books that start with an understanding of radians, and the unit circle, and others that start with the association of sides of triangles to a given angle. This is made more obvious why from looking at the two accepted national standards NCTM and CCSS we have to different interpretation of what is important in the understanding. So when looking in a book like Algebra, Larson 2004 we have no introduction to trigonometry with radians and trig functions. Larson, starts the section with triangle association. This book either would be taking the approach of NCTM. It is important to note that NCTM standards do not leave out that trigonometry can be used as circular functions however the order is different. Tutoring Tutoring student-athletes on my free time has allowed me to relearn trigonometric functions and their basics. Using Swokowski, E. W., & Cole, J. A. (2008). Precalculus: Functions and Graphs, I introduced students to radian measure and the unit circle before seeing any work done with trigonometric functions. The students were able to understand how the functions relate to the unit circle better from learning radians first. Following this understanding we did work with right triangles. Then found more applications to trig functions and finished with inverse functions. More Extensive Trigonometric Functions Trigonometric functions have been used recently in my university levels in both the courses that I tutor for pre-calculus and the Geometry course I am taking. The functions are used as a transformation linear function that will rotate lines about the origin. A transformation that is done by adding the angles and multiplying the coefficients or radii of the circle that is being rotated about by the respective function, which is far from any use of trigonometric function that I have used before. As I have continued in mathematics I have seen trigonometric functions show up in other courses of study. I have seen recently that trigonometric function is actually a transformation of points about an angle of measure. We can see applications of trig in physics, astrology, music, and calculus. From physics we branch into optics, which is very heavy on the use of trigonometric functions to evaluate very small angles and their measure. There are trigonometric functions used in vectors, where trigonometric functions are used to show direction of a vector. These functions can be found in navigation, one of the origins of trigonometry. Harmonic motion, which contains the sine and cosine wave graphs, clearly uses the functions. Harmonics are typically expressed as moving about a circle in a courter clockwise direction, thus similar to a unit circle in turn we notice how trig functions are applicable here. And a section that I find to be much harder is trigonometric functions as series and sequences.  Conclusion After doing research, I wonder when did, and why do students still instantly think of triangles when hearing trigonometric functions? What work a teacher or the past of a student that would create such a connection to be bonded so strongly has done? As I tutor more and more at the level that involves trigonometric function introduction, I see that most of all the work done is to be used in triangles. When teaching, I constantly bring students back to understanding the unit circles and the values of sin within the unit circle to foster a growing in understanding. The association to trigonometry and the comprehension of the unit circle is conceptually a better way of learning for students, when companied by the understanding of right triangle trigonometry. The ideas that are later harped to triangles and angle degrees of triangles, is made solid from the trigonometrys starting point. Thus there is no single sided way to teach trigonometry, I have found that teaching radians and the unit circle followed by trigonometry with the use of radians, will strengthen a students understanding of right triangle trigonometry. A combination of these two applications will provide students the knowledge they need to understand and apply trigonometric functions to other aspects of math. References Hass, J. (2011). University Calculus, Early Transcendentals. Upper Saddle River, NJ: Pearson Education. Axler, S. J. (2009). Precalculus: a Prelude to Calculus. Hoboken, NJ: Wiley. Swokowski, E. W., & Cole, J. A. (2008). Precalculus: Functions and Graphs. Belmont, CA: Thomson / Brooks/Cole. Larson, R. (2007). Algebra and Trigonometry. Belmont, CA: Brooks/Cole Cengage Learning. (2006). Retrieved from: http://hyperphysics.phy-astr.gsu.edu/hbase/trgser.html Kupkova E. (2005). Radians Versus Degrees. Acta Didactica Universitatis Comenianae Mathematics, 5, 87-94. Retrived from  HYPERLINK "http://www.ddm.fmph.uniba.sk/ADUC/files/Issue5/06%20Kupkova.pdf" http://www.ddm.fmph.uniba.sk/ADUC/files/Issue5/06%20Kupkova.pdf Weber K. (2005). Students Understanding of Trigonometric Functions. Mathematics Education Research Journal, vol. 17, 3, 91-112. Larson, R. (2004). McDougal Littell Algebra 2. Evanston, IL: McDougal Littell. Joyce, J.E. (1997). Applications of Trigonometry. Worcester, MA: Clark University. Retrieved from  HYPERLINK "http://www.clarku.edu/~djoyce/trig/apps.html" http://www.clarku.edu/~djoyce/trig/apps.html Musser, G.L. & William, B.F. (1997). Mathematics for Elementary Teachers: a conceptual Approach. Uper Saddle River, NJ: Prentice Hall Figure 1: Oojih A positive learning experience and measureable progress  HYPERLINK "http://www.oojih.com/show/trigonometry/generalangle/" http://www.oojih.com/show/trigonometry/generalangle/ Figure2: The online science digest;  HYPERLINK "http://www.sciencedigest.org/UnitCircle.gif" http://www.sciencedigest.org/UnitCircle.gif Degrees and Radians.  HYPERLINK "http://www.ucl.ac.uk/mathematics/geomath/trignb/deg.html" http://www.ucl.ac.uk/mathematics/geomath/trignb/deg.html Appendix A Georgia Performance standards (website) Trigonometric Functions F TF Extend the domain of trigonometric functions using the unit circle MCC9 12.F.TF.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. MCC9 12.F.TF.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. MCC9 12.F.TF.3 (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for/3, /4 and/6, and use the unit circle to express the values of sine, cosine, and tangent for x,+ x, and 2 x in terms of their values for x, where x is any real number. MCC9 12.F.TF.4 (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. Appendix B Common core(website) Standards in this domain: Extend the domain of trigonometric functions using the unit circle. F-TF.1.Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. F-TF.2.Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. F-TF.3.(+) Use special triangles to determine geometrically the values of sine, cosine, tangent for /3, /4 and /6, and use the unit circle to express the values of sine, cosines, and tangent forx,  +x, and 2  xin terms of their values forx, wherexis any real number. F-TF.4.(+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. Model periodic phenomena with trigonometric functions. F-TF.5.Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.& F-TF.6.(+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed. F-TF.7.(+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.& Prove and apply trigonometric identities. F-TF.8.Prove the Pythagorean identity sin2() + cos2() = 1 and use it to find sin(), cos(), or tan() given sin(), cos(), or tan() and the quadrant of the angle. F-TF.9.(+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. Appendix C Grades 9 12: Standard 9: Trigonometry In grades 9 12, the mathematics curriculum should include the study of trigonometry so that all students can- apply trigonometry to problem situations involving triangles; explore periodic real-world phenomena using the sine and cosine functions; and so that, in addition, college-intending students can- understand the connection between trigonometric and circular functions; use circular functions to model periodic real-world phenomena; apply general graphing techniques to trigonometric functions; solve trigonometric equations and verify trigonometric identities; understand the connections between trigonometric functions and polar coordinates, complex numbers, and series. Focus Trigonometry has its origins in the study of triangle measurement. Many real-world problems, including those from the fields of navigation and surveying, require the solution of triangles. In addition, important mathematical topics, such as matrix representations of rotations, direction angles of vectors, polar coordinates, and trigonometric representations of complex numbers, require trigonometric ratios, further underscoring the connections between geometry and algebra. Natural generalizations of the ratios of right-angle trigonometry give rise to both trigonometric and circular functions. These functions, especially the sine and cosine, are mathematical models for many periodic real-world phenomena, such as uniform circular motion, temperature changes, biorhythms, sound waves, and tide variations. Although all students should explore data from such phenomena, college-intending students should identify and analyze the corresponding trigonometric models. These students should also study identities involving trigonometric expressions and inverses of trigonometric functions, together with their applications to the solution of equations and inequalities. 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