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Building The Unit Circle

Building The Unit Circle. By Michael Geiswhite ITC 525 Computers for Educators 2011 Summer Session I. Building The Unit Circle. History of Trigonometry History of the Unit Circle Important Triangles SOH CAH TOA Ratios Quadrants The Final Unit Circle Fun Facts Standards Credits.

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Building The Unit Circle

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  1. Building The Unit Circle By Michael Geiswhite ITC 525 Computers for Educators 2011 Summer Session I

  2. Building The Unit Circle • History of Trigonometry • History of the Unit Circle • Important Triangles • SOH CAH TOA • Ratios • Quadrants • The Final Unit Circle • Fun Facts • Standards • Credits

  3. History of Trigonometry • Hipparchus of Nicaea is known as the father of trigonometry. He compiled the first trigonometric tables to simplify the study of astronomy more than 2000 years ago. He paved the way for other mathematicians and astronomers using triangle ratios. • The term “trigonometry” itself emerged in the 16th century, although it derives from ancient Greek roots: “tri” (three), “gonos” (side), and “metros” (measure).

  4. History of the Unit Circle • The idea of dividing a circle into 360 equal pieces dates back to the sexagesimal (base 60) counting system of the ancient Sumerians. The appeal of 60 was that it was evenly divisible by so many numbers (2, 3, 4, 5, 6, 10, 12, 15, 20, and 30). Early astronomical calculations wedded the sexagesimal system to circles, and the rest is history.

  5. Important Triangles • 30-60-90 Triangle • 45-45-90 Triangle

  6. SOH CAH TOA Click here for a SOHCAHTOA song. • SOH CAH TOA is an acronym someone came up with to help us remember how to find the sine, cosine, and tangent values of an angle. We use this in forming the unit circle by using the 30-60-90 and 45-45-90 triangles and ratios. Click here for SOHCAHTOA practice

  7. Pop Quiz 1 Why was the number 60 used as the base for the degrees in a circle? • It was Hipparchus’ favorite number • It is divisible by a lot of numbers • It was the year they invented trigonometry

  8. CORRECT! You have a very promising future in mathematics. GOOD JOB!

  9. Sorry, your answer is incorrect. Brush up on the history slides and try again.

  10. Ratios • We’re going to use ratios and the two important triangles to build the unit circle. First, we need to remember the definition of ratios. A ratio is a quotient of two numbers or quantities. Also, since we’re building the UNIT circle, we need to remember that “UNIT” means ONE. So we’re going to make each of the hypotenuses of the important triangles equal to one.

  11. Ratios of Important Triangles • For the 30-60-90 triangle, we will need to divide each side by 2 so that they hypotenuse will have a length of 1. Therefore, we’re left with a triangle that looks like this:

  12. Ratios of Important Triangles • For the 45-45-90 triangle, we will need to divide each side by the square root of 2. We will then need to rationalize each denominator and we’ll end up with a triangle like this:

  13. Pop Quiz 2 Once we have the triangles with 1 for the hypotenuse, what is the side length opposite of 30°?

  14. CORRECT! YOUR’RE A TRIG MASTER!!! GREAT JOB!

  15. Sorry, your answer is incorrect. Read the slide about the ratios of important triangles again.

  16. Quadrants • The unit circle is drawn on the coordinate plane so just like the coordinate plane, we have four quadrants. Sine, cosine and tangent are positive in exactly two quadrants and negative in the other two quadrants. Sine corresponds to the y-values and cosine corresponds to the x-values. Tangent is a ratio of sine values to cosine values. • Sine is positive in quadrants 1&2 and negative in 3&4 • Cosine is positive in quadrants 1&4 and negative in 2&3 • Tangent is positive in quadrants 1&3 and negative in 2&4

  17. Final Unit Circle We can take the ratio versions of the 30-60-90 and 45-45-90 triangles and place them inside a circle with radius of one to create the final unit circle.

  18. Fun Facts • Trigonometry is everywhere in our lives even though you may not have heard of it or know how to use the sine, cosine or tangent functions. • For example the mathematics behind trigonometry is the same mathematics that enables us to store sound waves digitally on a CD. So when you’re burning your favorite songs onto a CD, technically you’re using trigonometry without even knowing it.

  19. Fun Facts Continued... • The sine and cosine wave (pictured below) are the waves that are running through the electrical circuit known as Alternating Current. So when you plug something into the wall, which most of us do on a daily basis, we are again using trigonometry.

  20. Standards • 2.10.11.A – Identify, create, and solve practical problems involving right triangles using the trigonometric functions and the Pythagorean Theorem. • 2.1.G.C – Use ratio and proportion to model relationships between quantities. • 1. Facilitate and Inspire Student Learning and Creativity • Teachers use their knowledge of subject matter, teaching and learning, and technology to facilitate experiences that advance student learning, creativity, and innovation in both face-to-face and virtual environments. Teachers: • a. promote, support, and model creative and innovative thinking and inventiveness. • b. engage students in exploring real-world issues and solving authentic problems using digital tools and resources. • c. promote student reflection using collaborative tools to reveal and clarify students' conceptual understanding and thinking, planning, and creative processes. • d. model collaborative knowledge construction by engaging in learning with students, colleagues, and others in face-to-face and virtual environments.

  21. Credits • http://en.wikipedia.org/wiki/Sexagesimal • http://www.hps.cam.ac.uk/starry/hipparchus.html • http://www.youtube.com/watch?v=NeIm_aSFd3I • http://whatis.techtarget.com/definition/0,,sid9_gci798217,00.html • http://www.mathwarehouse.com/trigonometry/sine-cosine-tangent-practice2.html

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