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[Index]

   Introduction Introduction
   What's the essence? Essence
   Memorizing Memorizing
   Understanding Understanding
   Summarizing Summarizing
   Examples Examples

Example 1
Example 2
Example 3
Example 4
Example 5
Example 6
Example 7
Example 8
General ideas

   Exercises Exercises
   Final thoughts Final thoughts
     
ready for the last one?
Page 1 | Page 2 | Page 3 | Page 4 | Page 5 | Proof(s)

Proofs of example 8:

Suppose that a, b, c, theta.gif (908 bytes), and phi.gif (942 bytes) are real numbers, with the following conditions: a2_b2n_0.gif (1540 bytes), tht_n_p_2kpi.gif (1673 bytes), k is any integer.

6_8101.gif (1956 bytes) (1)
6_8102.gif (1950 bytes) (2)

Prove that 6_8103.gif (1903 bytes).

First proof | Second proof



First proof:

(1) + (2) 6_8104.gif (3139 bytes) (3)         

that is,  6_8p01.gif (1803 bytes)

Dividing by 2,   6_8106.gif (3075 bytes) (4)         
Squaring the both sides,  6_8108.gif (3381 bytes) (5)         

Claim

6_8202.gif (2926 bytes) (6)         
(1) - (2)  6_8203.gif (2974 bytes) (7)         

6_8204.gif (3591 bytes)

6_8205.gif (3053 bytes) (8)         

From tht_n_p_2kpi.gif (1673 bytes), we have: 6_8208.gif (1550 bytes)
From equation (8), we have:

6_8209.gif (2399 bytes) (9)         

From (6), we have:

6_8301.gif (4286 bytes) (10)         

Squaring (9) and adding it to (10), we have:

From equations (5) and (6), we have:

6_8308.gif (2206 bytes)

Since a2_b2n_0.gif (1540 bytes),

6_8310.gif (1901 bytes)

Go up (430 bytes)

Second proof:

From (1) 6_8406.gif (2265 bytes) (3)         
From (2) 6_8407.gif (2261 bytes) (4)         

where alpha.gif (882 bytes) satisfies 6_8402.gif (1927 bytes), or 6_8403.gif (1159 bytes)(if a = 0).

(3) + (4) 6_8409.gif (3426 bytes)
that is, 6_8410.gif (3205 bytes)
  6_8p02.gif (1394 bytes)

Since a2_b2n_0.gif (1540 bytes), 6_8p03.gif (2811 bytes)

Squaring the both sides,  6_8412.gif (2878 bytes) (5)         

Claim:

6_8413.gif (1791 bytes) (6)         

(3) - (4)   6_8501.gif (3120 bytes)

Since a2_b2n_0.gif (1540 bytes), 6_8502.gif (2451 bytes)

that is,  6_8504.gif (2490 bytes) (7)         

Since tht_n_p_2kpi.gif (1673 bytes), 6_8505.gif (1552 bytes)

so  6_8506.gif (1723 bytes)

Since 6_8507.gif (2785 bytes), 6_8508.gif (1797 bytes).  Hence (6) holds. From equations (5) and (6) we have:

6_8509.gif (1884 bytes)


                                                                             
                                                                                                                                                                                                                                         
     
  

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LWR
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Copyright 1998 LWR, ThinkQuest team 17119. All rights reserved.