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   Final thoughts

4. Understanding the formulas

In section 2, we learned that the process of verifying an identity is also the process of reducing the three differences between the two sides in the identity. The three differences are difference of functions, difference of operations, and difference of angles. We also showed that the process of verifying an identity is the process of using some basic formulas to transform one side of the identity to the other side. Hence it is important to understand the formulas. We know that the goals of using formulas are reducing the three kinds of differences. We have to understand every formula very well. To do so, we need to consider every formula from the three differences. To prove trigonometric identities, only memorizing them is not enough. We need to understand the role of every formula in transforming the three differences.

For example, 4_401.gif (3269 bytes), remember that both sides of the equation are equal, so it's OK to use it from either side. If we use it from the right side to the left side, then we can see that sine and cosine are changing to cosine, which changes the function. Don't forget our idea--reduce the three differences of functions, operations, and angles. Also, from right side to left side, it changes the operation, from addition to multiplication by 1, and it changes the angles, alpha.gif (882 bytes) and beta.gif (942 bytes) to al_p_bet.gif (1116 bytes).

From left to right, the function is changed from cosine to cosine and sine. The operation changed from multiplication to addition. The angle al_p_bet.gif (1116 bytes) is changed from alpha.gif (882 bytes) and beta.gif (942 bytes).

Here is an example, s_a_c_a1.gif (1861 bytes). If we use it from left side to the right side, this formula changes the operation from addition to multiplication, and deletes the angle alpha.gif (882 bytes) and the functions sine and cosine. From the right side to the left, it changes the operation from multiplication to addition of squares and fabricates angle alpha.gif (882 bytes) and functions sine and cosine.

Moving one term from the left side to the right side in the formula s_a_c_a1.gif (1861 bytes), for example, we have 4_102.gif (1769 bytes). Then from the left to right side, this formula helps us to change function cosine to function sine, and changes the multiplication to the subtraction of squares. From the right to the left side, this formula changes function sine to function cosine, and changes the operation from subtraction to multiplication.


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