6.3.1 Trigonometric Functions of Real Numbers

Radians vs. Real Numbers The argument of a trig function can be a real number, radians, or degrees. Sin(2) real number, radian, or degree? Sin(2  ) real number, radian, or degree? Note: sin(2) ≠ sin(2  )

Unit Circle The unit circle defines the trig functions in terms of the dependent variable If we consider time in radians as our x-values we can consider the yt plane (similar to xy plane except t values for x axis)

Functions defined P(x,y) on the unit circle sin(t) = y cos(t) = x tan(t) = csc(t) = sec(t) = cot(t) =

Since these functions are defined by a circle… On a circle 180  or half a circle or  Produces a t-value of exact opposite value Similarly 360  or a whole circle or 2  Produces a t-value of the same value

This defines our functions as follows: P(t) = (x, y) P(t +  ) = (-x, -y) P(t -  ) = (-x, -y) P(-t) = (x, -y)

Find P(t +  ), P(t -  ), P(-t), and P(-t -  ) Given,Since P(-t -  ) = P(-(t +  )) P(t) P(t +  ), P(t -  )P(-t) P(-t -  )

Remember! P(x,y) = P(cos(t), sin(t)) Where t is the angle in radians

Find the Values of the Trigonometric Fucntions 10)a) -  sin(-  )= 0 cos(-  )= -1 tan(-  )= 0 csc(-  )= U sec(-  )= -1 cot(-  )= U 0  (-1, 0) (2  )

You Try! 10)b) 6  sin(6  )= cos(6  )= tan(6  )= csc(6  )= sec(6  )= cot(6  )= 0  (2  )

Homework p , 5-8, 9-15 odd