4.24.2 TrigonometricTrigonometric Function:Function: TheThe UnitUnit CircleCircle The Unit Circle A circle with radius of 1 Equation x2 + y2 = 1 ( 1,0 ) cos( θ ,θ sin) ( 0,1 ) (− 0,1 ) θ 0( ,− ) 1 The Unit Circle with Radian Measures 2π Do you remember 30º, 60º, 90º triangles? Now they are really! Important Do you remember 30º, 60º, 90º triangles? Now they are really important! Do you remember 30º, 60º, 90º triangles? 3 Cos30° = 2 1 Sin30° = 2 Do you remember 30º, 60º, 90º triangles? 1 Cos60° = 2 1 3 Sin60° = 3 2 θ 1 2 Do you remember 45º, 45º, 90º triangles? When the hypotenuse is 1 The legs are 2 2 2 1 Cos45° = 2 2 2 2 Sin45° = 2 2 2 Some common radian measurements These are the Degree expressed in Radians π π 30° = 90° = 6 2 π 180° =π 45° = 4 3π 270° = π 2 60° = 3 360° = π 2 The Unit Circle: Radian Measures and Coordinates 2π The Six Trig functions b adjacent Cosθ= = Sinθ c hypotenuse Tanθ = a opposite Cosθ Sinθ= = c hypotenuse a opposite Tanθ= = b adjacent θ 1 Cos→ →Sec cos 1 Sin→ →Csc sin 1 Tan→ →Cot tan 2π Lets find the Trig functions if θ = 3 Think where this angle is on the unit circle. 2π −1 ⎛ −1 3 ⎞ ⎛ ⎞ ⎜ ⎟ Cos⎜ ⎟ = ⎜ , ⎟ ⎝ 3 ⎠ 2 ⎝ 2 2 ⎠ ⎛ 2π ⎞ 3 2π Sin⎜ ⎟ = θ = ⎝ 3 ⎠ 2 3 3 ⎛ 2π ⎞ Tan⎜ ⎟ = 2 = −3 ⎝ 3 ⎠ −1 2 Sinθ Tanθ = Cosθ 2π Find the Trig functions of θ = 3 Think where this angle is on the unit circle. ⎛ 2π ⎞ −1 ⎛ 2π ⎞ − 2 Cos⎜ ⎟ = Sec⎜ ⎟ = =2 − ⎝ 3 ⎠ 2 ⎝ 3 ⎠ 1 ⎛ 2π ⎞ 3 Sin⎜ ⎟ = ⎛ 2π ⎞ 2 2 3 ⎝ 3 ⎠ 2 Csc⎜ ⎟= = ⎝ 3 ⎠ 3 3 3 ⎛ 2π ⎞ 2π −1 − 3 Tan⎜ ⎟ = 2 = −3 ⎛ ⎞ −1 Cot⎜ ⎟ = = ⎝ 3 ⎠ ⎝ 3 ⎠ 3 3 2 − π How about θ = 4 ⎛ −π ⎞ 2 Cos⎜ ⎟ = ⎛ 2 2 ⎞ ⎝ 4 ⎠ 2 ⎜ ⎟ ⎜ , ⎟ ⎝ 2 2 ⎠ ⎛ −π ⎞ − 2 Sin⎜ ⎟ = ⎝ 4 ⎠ 2 − π 2 θ = 4 ⎛ −π ⎞ Tan⎜ ⎟ = 2 =1 − ⎝ 4 ⎠ − 2 ⎛ 2 − 2 ⎞ ⎜ ⎟ ⎜ , ⎟ 2 ⎝ 2 2 ⎠ There are times when Tan or Cot does not exist. At what angles would this happen? There are times when Tan or Cot does not exist. π⎧ 3 π⎫ ⎨ , ⎬ ⎩ 2 2 ⎭ If think of the domain of the trig functions, there are some limits. Look at the unit circle. If x goes with Cos, then what are the possible of Cos? It is the same with Sin? Definition of a Periodic Function A function “f” is periodic if there exists a positive real number “ c” such that f(t + c) = f(t) for all values of “t”. The smallest “c” is called the period. Even Function ( Trig. ) Cos (- t) = Cos (t) and Sec( -t) = Sec (t) Also Sin(-t) = -sin (t) and Csc (-t) = - Csc (t) Tan(-t) = -Tan (t) and Cot(-t) = - Cot (t).