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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

6_3101.gif (3874 bytes)

We don't know if you already noticed that this is a hard one. Before we do anything else, let's find out what are the differences between the two sides. The angles and operations are different. Let's look at it closely. The right side is tan_3tht.gif (1252 bytes). If you did read our Double & Triple Angle formulas in Memorizing the formulas section, you should know it equals to 6_3102.gif (1994 bytes). Believe it or not, you need to memorize it, or you will be in trouble, at least in this problem. Now the right side looks like this:

6_3103.gif (2443 bytes)

We will not say we can't, but we think it's not a good idea to prove it from the right side to the left side.

Once we know the right side, we need to compare both sides again. Now, we see that they have two differences. It didn't change a lot, but, it's very easy to change tht+120.gif (1502 bytes) to theta.gif (908 bytes) and 120.gif (1133 bytes), and 120.gif (1133 bytes) is special angle, the tangent of which could be changed to a special number. So we use Sum formulas to do this.

6_3104.gif (4431 bytes)

Now, we want to change the values of functions of special angles to special numbers, so we leave tan_tht.gif (359 bytes) there.

6_3105.gif (3587 bytes)

We know that the right side is multiplication, so we want to change the operations of left side to multiplication too. If we simplify the left side, we get:

6_3106.gif (10642 bytes)


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