Chapter 12 Trig Identities by wuzhengqin

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									Chapter 12 Trig Identities
A Trig identity is an equation that is true for all values of the variable.

A Trig equation is an equation that is true for only certain values.

Trig Identities are useful in several ways – for simplifying trig expressions, proving other
trig identities, evaluating trig functions, and solving trig equations

      Reciprocal Identities            Quotient Identities           Pythagorean Identities
                    1                             sin 
          csc                          tan                             cos 2   sin 2   1
                  sin                           cos 
                       1                             cos 
           sec                           cot                           1  tan 2   sec 2 
                     cos                            sin 
                       1
           cot                                                           cot 2   1  csc2 
                     tan 

       Basic Identity                                Alternative Forms
      cos2 θ + sin2 θ = 1           cos2 θ = 1 – sin2 θ            sin2 θ = 1 – cos2 θ
                   1                                                              1
         sec                       sec θ cos θ = 1                    cos 
                 cos                                                           sec
                   1                                                              1
         csc                        csc θ sin θ = 1                   sin  
                 sin                                                           csc
                   1                                                              1
         cot                        cot θ tan θ = 1                   tan  
                 tan                                                           cot 

Use Trig Identities to simplify the following expressions:

                                                                            sec x  cos x
(a) sin x cos x sec x         (b) cos2  1  tan 2                (c)
                                                                                tan x




       tan x                        1  cot A
(d)                           (e)                            (f) cos3 x  sin 2 x cos x
      sec( x)                         csc A
                                 Proving Trig Identities
To prove an equation is an identity, show that both sides of the equation can be written
in the same form, that is, you see the same thing on both sides, just like when we check
equations. To do this, use valid substitutions and operations and follow the following
procedures:

PROCEDURE:
  1. Transform the expression on one side of the equality (usually the more
     complicated expression) into the form of the other side (make one side look like
     the other.)
  2. Transform both sides of the equality into a form common between the two sides
     When in doubt, change everything to sin & cos. (work on one side at a time.)
  3   Simplify complex fractions. Look for a common factors in the numerator &
     denominator in order to reduce the fraction to lowest terms

***NOTE – It is incorrect when proving identities to add/subtract, multiply or
divide terms across the = sign. All substitutions or algebraic operations must
take place on one side of the = or the other. You cannot transfer terms across the
=. Prove an identity in the same manner you demonstrate a check.

Simple Substitution                            Products:
Prove: tan 2   sin 2   cos2   sec2      Prove: cos θ(sec θ – cos θ) = sin2 θ




Factor:
           sin 2 
Prove:               1  cos 
          1  cos 




Adding Fractions: (need a __________ _______________!)
                      cos   1
Prove: 1  sec                               Prove: tan θ + cot θ = sec θ csc θ
                       cos 
Simplifying Fractions:

         sec 
Prove:          tan 
         csc 




Prove the following identities:
1  csc 2  1  sin 
                                      sin A  cos A cot A  csc A
1  csc      sin 




                                       sec   cos 
(sin x  cos x) 2  1  2sin x cos x                  sin 2 
                                           sec 
   Cosine of the Sum of Two Angles                              Cosine of the Difference of Two Angles

cos (A + B) = cos A cos B – sin A sin B                         cos (A – B) = cos A cos B + sin A sin B
                                                  3                                B
   1. If x and y are acute angles and sin x =       , and   2. Using the formula for cos (x + y), prove
                                                  5
             12
   cos y =      , find cos (x + y)                          cos (π + A) = –cos A.
             13




                                     4
   3. If A is in QI with cos A =      and B is in
                                                                                                   o
                                                            4. Find the exact value of cos 75 using
                                     5
                          5
   QII with sin B = 
                                                                    o      o
                            , find cos (A - B).             cos (30 + 45 ).
                         13




                                   12                                                        3
   5. If x is acute and cos x =       , find cos (π – x)    6. . If x is acute and cos x =     and y is obtuse
                                   13                                                        5
                                                                           5
                                                            and sin y =      , find cos (x – y).
                                                                          13




                     o
   7. prove cos(90 – x) = sin x                             8. If cos ( A – 30) = ½ , find measure of m     A
     Functions of the Sum of Two Angles                         Functions of the Difference of Two Angles
     sin (A + B) = sin A cos B + cos A sin B                    sin (A – B) = sin A cos B – cos A sin B


                       o      o                                               o             o   o
1. Find the sin (60 + 30 )                              2. Find the sine 120 using sin (180 - 60 )




                                                        4. Find the sin (    )
                   o
3. Prive sin (90       – x) = cos x




                  4            4
5. . If sin x =     and cos y = and x & y are measures of angles in the first quadrant, find the value of
                  5            5
sin ( x + y)




                                                                   3                   5
6.   Simplify the expression sin   x                 7. If sin A =    and cos B = 
                                 6                                  5                  13
                                                           Find sin ( A + B) and sin ( A - B)
                     Tangent of the Sum & Difference of 2 Angles

               SIN (A+B)    sin A cos B +cos A sin B     tanA + tan B
TAN(A + B) =              =                            =
               COS( AB )    cos A cos B  sin A sin B   1  tan A tan B


               SIN (A-B)    sin A cos B -cos A sin B      tanA - tan B
TAN(A - B) =              =                            =
               COS( AB )    cos A cos B  sin A sin B   1  tan A tan B



   1. Show that tan 120o =  3 by using the formula tan ( 60 o + 60 o)




   2. Show that tan 90o is undefined using the formula tan ( 60 o + 30 o)




   3. Use the tangent of the sum of two angles to prove: tan (180 o + x) = tan x




                   5            1
   4. If tan A =     and tan B = , find tan(A - B)
                   4            5
                                                              1  tan x
5. Use the tan (A + B) identity to prove: tan (315o + x) =
                                                              1  tan x




6. Find tan (2   )




7. If tan x = -6 and tan y = ½ find tan (x + y) and tan (x – y)




8. Find the exact value of tan 15 o using tan (45 o – 30 o)




9. find the exact value of tan 195 o using tan (135 o + 60 o)
Double angles and ½ angle formulas.
     Functions of the Double Angle                                    Functions of the Half Angle
    sin 2A = 2 sin A cos A                                    1  cos A                      1  cos A
                                                sin 1 A                  cos 1 A  
    cos 2A = cos2 A – sin2 A                        2
                                                                  2            2
                                                                                                 2
    cos 2A = 2 cos2 A – 1
                                                     1        1  cos A 1  cos A       sin A
    cos 2A = 1 – 2 sin2 A                       tan    A                         
                2 tan A                              2        1  cos A     sin A     1  cos A
    tan 2 A                                        o          o                    o
                                                If 0 ≤ x ≤ 180 , then 0 ≤ 2 x ≤ 90 and 1 x is in Q___.
                                                                           1
              1  tan 2 A                              o           o        o              o
                                                                                            2
                                                If 180 ≤ x ≤ 360 , then 90 ≤ 2 x ≤ 180 and 1 x is in Q___.
                                                                                1
                                                                                                2
                                                       o           o          o               o
                                                If 360 ≤ x ≤ 540 , then 180 ≤ 2 x ≤ 270 and 1 x is in Q___.
                                                                                  1
                                                                                                  2
                                                If 540o ≤ x ≤ 720o, then 270o ≤ 1 x ≤ 360o and 1 x is in Q___.
                                                                                  2               2




1. If cos A =  13 and A is an  in QII, find
                 5
                                                      2. If x is acute and cos x =       3
                                                                                         5
                                                                                             , find sin   1
                                                                                                          2
                                                                                                              x.
  a. sin 2A. b. cos 2A.




3. If sin x = –0.8 and x is in QIV, find              4. If cos A =  14 and A is in QIII, find cos
                                                                      64
                                                                                                                   1
                                                                                                                   2
                                                                                                                       A.
  a. cos 2x                   b. sin 2x.




5. If cos A =  5 and A is in QIII, find
                3
                                                      6. If θ is in QIV and sin θ =  5 , find sin  .
                                                                                      4
                                                                                                   2
  a. cos 2A                   b. sin 2A
                                                                                      3               1
7. If cos x = a, express cos 2x in terms of a.          8. If x is obtuse and sin x = 5 , find cos    2
                                                                                                          x.




                                           7                             1                           4
9. if x is an acute angle and cos x =                   10.Find cos        B if B is acute & Tan B =
                                          32                             2                           3
           x
Find sin
           2




     Reciprocal Identities                     Quotient Identities                  Pythagorean Identities
                 1                                      sin 
       csc                                    tan                                cos 2   sin 2   1
               sin                                     cos 
                 1                                      cos 
       sec                                    cot                                1  tan 2   sec 2 
               cos                                     sin 
                 1
       cot                                                                         cot 2   1  csc2 
               tan 

                  Sum of  Measures                                  Difference of  Measures
    sin (A + B) = sin A cos B + cos A sin B               sin (A – B) = sin A cos B – cos A sin B
    cos (A + B) = cos A cos B – sin A sin B               cos (A – B) = cos A cos B + sin A sin B
                Double-Angle Measures                                   Half-Angle Measures
                sin 2A = 2 sin A cos A
               cos 2A = cos2 A – sin2 A                              1  cos A                     1  cos A
                                                      sin 1 A                      cos 1 A  
                 cos 2A = 1 – 2 sin2 A                    2
                                                                         2               2
                                                                                                       2
                cos 2A = 2 cos2 A – 1                                       1     1  cos A
                              2 tan A                                 tan     A
                  tan 2 A                                                  2     1  cos A
                            1  tan 2 A
Let’s take a final look at proofs. Notice that we can use the double angle and half angle formulas
in our proofs. They can make a proof easier because they are easier to determine what to replace:

                                               sin 2               cos 2
sin 2θ sec θ = 2 sin θ               sin                                  sin   csc   sin 
                                               2 cos                sin 




                                                                 cos 2 A  cos A  1
(cos   sin  )2  1  sin 2                                                        cot A
                                                                   sin 2 A  sin A

								
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