PreCalculus: Lesson Plan * 8/18/2006

• Quickstart: Evaluate the six trig functions for 4π/3
• Lesson
  (4.2) Trig Functions: The Unit Circle
  
  1. Evaluate Trig Functions
     Use the unit circle & the definition of trig functions to evaluate the trig functions. Find the angle, t, (or its coterminal angle) on the unit circle. Note the (x, y) coordinates at that point. Use the definition of the trig functions to determine the value of the function at that point.
     Example: Evaluate the six trig functions at t = 5π/2
     Solution: t = 5π/2 is not on the unit circle, but we can find its coterminal angle by moving counterclockwise around the unit circle one and a quarter revolutions or by taking 5π/2 - 2π. It corresponds to π/2 and the point (x, y) = (0, 1).
     Therefore,
     sin(5π/2) = y = 1
     csc(5π/2) = 1/y = 1
     cos(5π/2) = x = 0
     sec(5π/2) = 1 / x undefined (1/0)
     tan(5π/2) = y/x undefined (1/0)
     cot(5π/2) = 0/1 = 0
     Example: Evaluate the six trig functions at t = -π/3 (Example 3 p.298)
     Solution: Moving clockwise around the unit circle (because t is negative), it follows that t = -π/3 is coterminal to 5π/3 and corresponds to the point (x, y) = (1/2, -sqrt(3)/2). Use definition of the trig functions to evaluate.
  2. Hints to memorize the unit circle
     Alphabetical order
     tan t = y / x similar to slope = (y2 - y1)/(x2 - x1)
     Ed's Method
     ALL Students Take Calculus - In Quad I ALL of the functions are positive, in Quad II Sine is positive, in Quad III Tangent is positive, in Quad IV Cosine is positive.
  3. Domain and Range of Sine and Cosine
     Domain for both is all Real Numbers [the input can be negative (-π/3), decimal (2.25), positive(π), fraction(5π/2), more than 2π]
     The range for both is [-1, 1]. Looking at the coordinates of the points on the outside of the Unit Circle the x-coordinate (corresponds to cosine) are not less than -1 and not more than 1. The y-coordinates (corresponds to sine) are not less than -2 and not more than 1.
  4. Even and Odd Functions
     Cosine (and its reciprocal secant) is an even function because cos t = cos (-t).
     Sine (and its reciprocal cosecant) is an odd function because sin (-t) = -sin t.
     Is tangent (and its reciprocal cotangent) an even or an odd function?
     

• HW: Study Unit Circle - Quiz Monday - you will be given a Blank Unit Circle and asked to fill it in with radian measures, degree measures, x- and y-coordinates.

Block I
All Student Code of Conduct acknowledgement forms were due today.

Block III
Reading Comprehension Check #1 collected today.
Create a vocabulary quiz for a friend using 10 words from your novel.