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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
     
a better way
Page 1 | Page 2 | Page 3 | Proof(s)

Example 4

6_4101.gif (3859 bytes)

We chose this identity as an example because it appears in a lot of trigonometry books, and it is a good problem.

Just as usual, we start from left to right. Let's see what are the differences. This identity has all three differences. We can tell you there are several ways to prove this problem. First let's see the easiest way.

First we want to reduce the differences of operations. Arrange the four terms on the left into two groups. Try to make them become products. The first two terms 6_4102.gif (1353 bytes) is actually 6_4103.gif (1335 bytes), by changing around 6_4104.gif (1924 bytes). The last 2 terms on the left side is 6_4105.gif (1666 bytes), which can also be converted to 6_4106.gif (1945 bytes)6_4107.gif (1629 bytes). Now we get:

6_4108.gif (2372 bytes)

When we look at the right side, we see that we already got something we need at the left side, which is 2sin_tht.gif (1209 bytes). The operation on the right is multiplication. So we factor 2sin_tht.gif (1209 bytes) out, and get:

6_4109.gif (2356 bytes)

By using our compare strategy, we can see that if we pull out 2sin_tht.gif (1209 bytes) from right side, which is 6_4110.gif (2237 bytes), then because both sides are equal, 6_4111.gif (1600 bytes) should equal to 6_4112.gif (1726 bytes). If we use the Sum-to-Product formulas to reduce the difference of operations, we have:

6_4113.gif (4437 bytes)

The identity has been proven.

The first way is the easiest and most natural way. But, by using more than one way to prove an identity, you can increase your problem solving skill tremendously. It can also help to "span" your brain faster.

     
  

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