Trig Functions of Real Numbers Lesson 2.4a

2 The Unit Circle  Consider a circle with radius r = 1  Wrap t onto the circumference  Then w(t) is a function which wraps t to a point P(x, y)  Also, t translates to θ (radians) t θ P(x, y) r = 1

3 The Unit Circle  The trig ratios for θ can tell us x and y Since r = 1 θ P(x, y) r = 1 View Nspire Demo View Nspire Demo

4 Example  Given  Then What are sin, cos, tan?  What if Determine P(x, y) What are the trig functions? x y

5 Implications  It is now possible to take functions of angles greater than 360 (2 π ) or less than -360 (-2 π )  Try these  Use both Wrapping concept Calculator (watch angle mode)

6 Properties of Trig Functions  Odd functions f(-x) = - f(x)  Even functions f(-x) = f(x)  Which of the trig functions are?? Odd Even This definition is also applied to non trig functions.

7 Trig Functions are Periodic  The functions repeat themselves  The period is the smallest value, p for which f(x) = f(x + p)  For sin, cos, sec, csc The period is 2 π  For tan and ctn The period is π

8 Assignment  Lesson 2.4a  Page 166  Exercises 1 – 47 odd