Trigonometric Ratios in Right Triangles

Trigonometric Ratios in Right Triangles M. Bruley

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

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

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:

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

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.

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

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.

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?

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)

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.

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.

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.

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

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

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

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

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.)

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

Trigonometric Ratios in Right Triangles