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Trigonometric Ratios in Right Triangles

Trigonometric Ratios in Right Triangles. Geometry Mr. Oraze. Trigonometric Ratios are based on the Concept of Similar Triangles!. 1. 45 º. 2. 1. 1. 45 º. 2. 45 º. All 45º- 45º- 90º Triangles are Similar!. 30º. 30º. 2. 60º. 60º. 1. 30º. 60º.

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Trigonometric Ratios in Right Triangles

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  1. Trigonometric Ratios in Right Triangles Geometry Mr. Oraze

  2. Trigonometric Ratios are based on the Concept of Similar Triangles!

  3. 1 45 º 2 1 1 45 º 2 45 º All 45º- 45º- 90º Triangles are Similar!

  4. 30º 30º 2 60º 60º 1 30º 60º All 30º- 60º- 90ºTriangles are Similar! 4 2 1 ½

  5. All 30º- 60º- 90ºTriangles are Similar! 10 60º 2 60º 5 1 30º 30º 1 60º 30º

  6. c a b The ratio is called the Tangent Ratio for angle  The Tangent Ratio c’ a’   b’ If two triangles are similar, then it is also true that:

  7. Side Opposite q Side Adjacent q Naming Sides of Right Triangles Hypotenuse q

  8. Tangent q = Hypotenuse Side Opposite q q Side Adjacent q The Tangent Ratio There are a total of six ratios that can be made with the three sides. Each has a specific name.

  9. Hypotenuse Side Opposite q q Side Adjacent q The Six Trigonometric Ratios(The SOHCAHTOA model) S O H C A H T O A

  10. Hypotenuse Side Opposite q q Side Adjacent q The Six Trigonometric Ratios The Cosecant, Secant, and Cotangent of q are the Reciprocals of the Sine, Cosine,and Tangent of q.

  11. 2 1 Solving a Problem withthe Tangent Ratio We know the angle and the side adjacent to 60º. We want to know the opposite side. Use the tangent ratio: h = ? 60º 53 ft Why?

  12. y x q Trigonometric Functions on a Rectangular Coordinate System Pick a point on the terminal ray and drop a perpendicular to the x-axis. (The Rectangular Coordinate Model)

  13. y x q Trigonometric Functions on a Rectangular Coordinate System Pick a point on the terminal ray and drop a perpendicular to the x-axis. r y x The adjacent side is x The opposite side is y The hypotenuse is labeled r This is called a REFERENCE TRIANGLE.

  14. y r y x q x Trigonometric Values for angles in Quadrants II, III and IV Pick a point on the terminal ray and drop a perpendicular to the x-axis.

  15. y q x Trigonometric Values for angles in Quadrants II, III and IV Pick a point on the terminal ray and raisea perpendicular to the x-axis.

  16. y q x Trigonometric Values for angles in Quadrants II, III and IV Pick a point on the terminal ray and raise a perpendicular to the x-axis. x y r Important! The  is ALWAYS drawn to the x-axis

  17. y x Signs of Trigonometric Functions Sin (& csc) are positive in QII All are positive in QI Tan (& cot) are positive in QIII Cos (& sec) are positive in QIV

  18. y x Signs of Trigonometric Functions Students All Take Calculus is a good way to remember!

  19. y (0, 1) x 90º Trigonometric Values for Quadrantal Angles (0º, 90º, 180º and 270º) x = 0 y = 1 r = 1 Pick a point one unit from the Origin. r

  20. 1 45 º 1 For Reciprocal Ratios, use the facts: Trigonometric Ratios may be found by: Using ratios of special triangles For angles other than 45º, 30º, 60º or Quadrantal angles, you will need to use a calculator. (Set it in Degree Mode for now.)

  21. Acknowledgements • This presentation was made possible by training and equipment provided by an Access to Technology grant from Merced College. • Thank you to Marguerite Smith for the model. • Textbooks consulted were: • Trigonometry Fourth Edition by Larson & Hostetler • Analytic Trigonometry with Applications Seventh Edition by Barnett, Ziegler & Byleen

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