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Trigonometric Ratios in Right Triangles M. Bruley Trigonometric Ratios are based on the Concept of Similar Triangles! All 45º- 45º- 90º Triangles are Similar! 2 45 º 1 1 45 º 1 2 1 2 1 2 2 45 º 2 2 All 30º- 60º- 90º Triangles are Similar! 2 30º 3 30º 60º 4 2 3 1 60º 1 30º 60º ½ 2 3 2 All 30º- 60º- 90º Triangles are Similar! 2 60º 10 1 30º 30º 3 5 3 1 60º 30º 3 2 1 2 60º 5 The Tangent Ratio c c’ a b a’ b’ a a' If two triangles are similar, then it is also true that: b b' The ratio a is called the Tangent Ratio for angle b Naming Sides of Right Triangles The Tangent Ratio Tangent Opposite Adjacent There are 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) Opposite Sine θ Hypotenuse Adjacent Cosine θ Hypotenuse Opposite Tangent θ Adjacent The Six Trigonometric Ratios Opposite Hypotenuse Adjacent Cosine θ Hypotenuse Opposite Tangent θ Adjacent Sine θ Hypotenuse Opposite Hypotenuse Secantθ Adjacent Adjacent Cotangent θ Opposite Cosecantθ 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 3 60º 53 ft 1 tan 60 opp h adj 53 3 h Why? 1 53 h 53 3 92 ft Cofunctions p. 287 There are three pairs of cofunctions: `The sine and the cosine The secant and the cosecant The tangent and the cotangent 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