PreCalculus: Lesson Plan * 9/18/2006

• Quickstart p.307 #19, 23
• Lesson
  Chapter 5 Analytic Trigonometry
  (5.1) Using Fundamental Identities
  Section Objectives: Students will know how to use fundamental trigonometric identities to evaluate trigonometric functions and simplify trigonometric expressions.
  
  1. Introduction (p. 376)
     Review the following list of identities that we have covered so far.
     Reciprocal Identities
     sin u = 1 / csc u
     cos u = 1 / secu
     tan u = 1 / cot u
     csc u = 1 / sin u
     sec u = 1 / cosu
     cot u = 1 / tan u
     Quotient Identities
     tan u = sin u/cosu
     cot u = cos u/sin u
     Pythagorean Identities
     sin²u + cos²u = 1
     tan²u + 1 = sec²u
     1 + cot²u = csc²u .
     Cofunctions Identities
     sin (90° - u) = cos u
     tan (90° - u) = cot u
     sec (90° - u) = csc u
     cos (90° - u) = sin u
     cot (90° - u) = tan u
     csc (90° - u) = sec u
     Even/Odd Identities
     sin (-u) = - sin u
     cos (-u) = cos u
     tan (-u) = - tan u
     cot (-u) = - cot u
     sec (-u) = sec u
     csc (-u) = - csc u
     
  2. Using the Fundamental Identities (pp 377 - 380)
     
     • Example 1. If csc u = -5/3 and cos u > 0, find the values of the other five trigonometric functions.
       sin u = -3/5
       cos²u = 1- sin²u = 1- (-3/5)² = 16/25
       cos u = 4/5
       sec u = 5/4
       tan u = sin u/cosu = (-3/5) / (4/5) = -3/4
       cot u = -4/3
     • Example 2. Simplify the following
     1. csc²x cot x – cot x
        = (csc² x – 1)cot x = (cot² x)cot x = cot³x
     2. tan x sin x + cos x
        = (sin x / cos x)sin x + cos x
        = (sin²x/cosx) + cosx
        = (sin²x/cosx) + (cos²x/cosx)
        = (sin²x + cos²x)/cos x
        = 1/cos x
        = sec x
     3. (sec t/tan t) - (tant/1+ sec t)
        = [sec t(1+ sec t)- tan²t]/[tant(1+ sec t)]
        = [sec t + sec²t - tan²t]/tan t(1+ sec t)
        = (sec t + 1)/[tant(1 + sec t)]
        = 1/tan t
        = cot t
        
     • Example 3. Factor the following trigonometric expressions
     1. cos²x – 1
        = (cos x + 1)(cos x – 1)
        b) sin²u – 3sin u – 10
        = (sin u + 2)(sin u – 5)
        c) sec²t – tan t – 3
        = (tan²t + 1) – tan t – 3
        = tan²t – tan t – 2
        = (tan t + 1)(tan t – 2)
        
     • Example 4. Rewrite 1/(sec x -1) so that it is not a fraction.
       1/(sec x -1) = [1/(sec x -1)] × [(sec x + 1)/(sec x + 1)]
       = (sec x + 1)/(sec² x - 1)
       = (sec x +1)/tan ²x
       = (sec x/tan² x) + (1/tan²x)
       = [(1/cosx)(1/tanx)(1/tanx)] + (1/tan²x)
       = [(1/cosx)(cosx/sinx)(1/tanx)] + (1/tan²x)
       = [(1/sinx)(1/tanx)] + (1/tan²x)
       = cot x csc x + cot²x
       
     • Example 5. Use the substitution x = 3sin u, 0 < u < p/2,
       to express √(9 - x²) as a function of u.
       √(9 - x²)
       = √(9 -(3sin u)²)
       = √(9 - 9sin² u)
       = 3 √(1- sin² u)
       = 3 √(cos² u)
       = 3cosu
       
• Homework p.381 - 383 #1 - 70 x 3's (Due Thursday)