Home Normal Version
IE Version
Search | BBS | Guest Book | Site Map | Help | Contact us | About us
Light version



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

   Final thoughts
      Page 1 | Page 2 | Page 3 | Page 4 | Proof(s)

Example 3--Page 3

Now, how do we combine the 2nd and 3rd terms together? Well, we all know that 6_3301.gif (1887 bytes). So we want to change tangent to sine and cosine.

6_3302.gif (4257 bytes)

Then we combine the last 2 terms.

6_3303.gif (5724 bytes)

Please note that the top of the fraction actually is 6_3304.gif (2398 bytes) by the Sum and Difference formulas. This might be the trickiest part of this proof, but it is still natural, because Sum formulas can help us to make sum to product--two terms combine to one term.

6_3305.gif (3443 bytes)

From some basic definition and the Derived formulas we know that value of the function of an angle plus 360.gif (1118 bytes) actually is same as the value of the function of original angle. So we have:

6_3306.gif (3064 bytes)

So what we do next? Well, good question. You can go ahead and take 10 seconds to think about what we do next.

OK, let me make this clear, what is our goal? We want to make the top become sin_3tht.gif (1224 bytes), bottom become cos_3tht.gif (1208 bytes). Genie, I wish... No, no, no, it won't work. We need to think by ourselves. We already got angle 2theta.gif (991 bytes) at the top, but we still want to change the bottom to angle 2theta.gif (991 bytes), which will be close to 3theta.gif (1006 bytes). So do you want to expand 6_3307.gif (1696 bytes) and 6_3308.gif (1656 bytes) by using Sum formula? No, don't do that. Since we will not get angle 2theta.gif (991 bytes). Let's see, does 6_3307.gif (1696 bytes)6_3308.gif (1656 bytes) looks familiar? No? How about this: cos_alph.gif (1101 bytes)cos_beta.gif (1131 bytes). Yes, it is one of the Product-to-Sum formulas. cos_alph.gif (1101 bytes)cos_beta.gif (1131 bytes)6_3309.gif (2632 bytes), and if we use this formula we will get angle 2theta.gif (991 bytes).

6_3310.gif (5106 bytes)


Previous | Top | Next


Questions? Comments? Contact us.
Copyright 1998 LWR, ThinkQuest team 17119. All rights reserved.