Introduction to Trigonometry

What is Trigonometry? Trigonometry is the study of how the sides and angles of a triangle are related to each other. It's all about triangles!

A Right Triangle Opposite Hypotenuse Adjacent

Same Right Triangle – Different Angle Hypotenuse B Adjacent Opposite

Trig Definitions: Sine = opposite/hypotenuse Cosine = adjacent/hypotenuse Tangent = opposite/adjacent Cosecant = hypotenuse/opposite Secant = hypotenuse/adjacent Cotangent = adjacent/opposite

A Way To Remember Sin = Opposite/Hypotenuse Oprah Had Cos = Adjacent/Hypotenuse A Huge Tan = Opposite/Adjacent Old Afro

y x y x r O x,y r = x + y 222

Definitions of Trig Functions Sin =y / r Cos =x / r Tan =y / x Csc=r / y Sec=r / x Cot= x / y O O O O O O

Radius = 1 The Unit Circle

y x 1/2 30 x,y r = x + y √3/2

y x √/2/2 45 x,y r = x + y √2/2

y x 1/2 60 x,y r = x + y √3/2

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)

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.

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

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. 

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

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

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

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

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:

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