Trigonometric Ratios in Right Triangles M Bruley Trigonometric

Trigonometric Ratios in Right Triangles M. Bruley

Trigonometric Ratios are based on the Concept of Similar Triangles!

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

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

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

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

Naming Sides of Right Triangles

The Tangent Ratio Tangent = There a total of six ratios that can be made with the three sides. Each has a specific name.

The Six Trigonometric Ratios (The SOHCAHTOA model)

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

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

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

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

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

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

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

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

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

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

Trigonometric Ratios may be found by: Using ratios of special triangles 1 45 º 1 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:

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