1. Trigonometric Ratios in Right Triangles M. Bruley

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

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

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

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

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

7. Naming Sides of Right Triangles   

8. The Tangent Ratio   Tangent  There are a total of six ratios that can be made with the three sides. Each has a specific name.

9. The Six Trigonometric Ratios (The SOHCAHTOA model)  

10. The Six Trigonometric Ratios   The Cosecant, Secant, and Cotangent of  are the Reciprocals of the Sine, Cosine,and Tangent of 

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

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

13. Trigonometric Functions on a Rectangular Coordinate System x y  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. Trigonometric Values for angles in Quadrants II, III and IV x y Pick a point on the terminal ray and drop a perpendicular to the x-axis.  y x r

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

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

17. Signs of Trigonometric Functions x y A A ll are positive in QI T Tan (& cot) are positive in QIII S S in (& csc) are positive in QII C Cos (& sec) are positive in QIV

18. Signs of Trigonometric Functions x y A A ll T Take S S tudents C Calculus is a good way to remember!

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

20. Trigonometric Ratios may be found by: 45 º 1 1 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.) For Reciprocal Ratios, use the facts:

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