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   Introduction Introduction
   What's the essence? Essence
   Memorizing Memorizing

  Basic Formulas
  Sum and Difference Formulas
  Double & Triple Angle Formulas
  Half Angle Formulas
  Product to Sum Formulas
  Sum to Product Formulas
  All Formulas

   Understanding Understanding
   Summarizing Summarizing
   Examples Examples
   Exercises Exercises
   Final thoughts Final thoughts

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Double-Angle Formulas:

3_201.gif (8281 bytes)

In Sum and Difference formulas, if alpha.gif (882 bytes) and beta.gif (934 bytes) are the same angle, which we let them equal to theta.gif (900 bytes), then you get these Double Angle formulas. Also, a 2 always appears in the formulas, as a coefficient 2 or as a square. The term cos_2tht.gif (1208 bytes) is a term that shows up a lot (3 times). Being strong in these formulas will make solving identities a lot easier. The term sin_2tht.gif (1232 bytes) is also common in real problems.

3_202.gif (2281 bytes)

Formula 1, the product of mixed functions sine and cosine, just don't forget the coefficient 2.

3_203.gif (2392 bytes)

Formula 2, remember that on the right side cosine and sine are all squared and it is the square of cosine minus the square of sine. Be aware of the order of the subtraction. Try to compare with sin_cos1.gif (1861 bytes).

3_204.gif (3558 bytes)

Formulas 3 and 4 come from formula 2 by using sin_cos1.gif (1861 bytes). Notice the order of the terms: cos_2tht.gif (1208 bytes) equals to 3_205.gif (1460 bytes)but not 3_206.gif (1440 bytes); 3_207.gif (1460 bytes) but not 3_208.gif (1467 bytes). Later, in Understanding the Formulas we will talk more about this.

3_209.gif (2356 bytes)

Formula 5, from the Sum formulas of tangent we can easily get this one when angle alpha.gif (882 bytes) = beta.gif (934 bytes)= theta.gif (900 bytes).

3_210.gif (3823 bytes)

There is also one more thing we want to mention, which is the Triple Angle formulas. They are not in most books, but we want to let you know them, because sometime you will need them to prove identities.

Triple Angle Formulas (5968 bytes)


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