MATH122

Is there a distributive law for trig functions?

This is a major DANGER area!

Example 1: True or false? Is the same as ????

Let’s evaluate both expressions using steps that we know are legal. We know that parenthesis indicate that that operation is to be completed first, so we can add and , then evaluate the one trig function that results.

Now for the second expression: there is no parenthesis, so we must evaluate each function separately, and then add the resulting values.

Argh! , so is NOT the same as !!!

We conclude that because “sin u” means “evaluate the sine function at u”, not “sine times u” that there is not a distributive law for applying trig functions to angles in general. We are so accustomed to using a distributive law when we see parentheses that the notation will often be misleading. We will have to work hard at recognizing the situation.

Example 2: Now you try it for cos (p + p) and cos p + cos p.

cos (p + p) = cos 2p = ?

cos p + cos p = ?

Is cos (p + p) the same as cos p + cos p?

Perhaps the quadrantal angles are special cases and we should try some less “special” values. How about this one?

Example 3: Is the same as ? You try it.

No, the “obvious” relationship between cos (u - v) and cos u, cos v does not exist, but there is a considerably more complicated one between cos (u - v) and cos u, cos v, sin u, sin v. It is a new trig identity for the cosine of the difference between two angles:

cos (u - v) = cos u cos v + sin u sin v

Example 4: Is ?

Yes! The two expressions are equal; the identity does indeed work!

Example 5: Evaluate cos 84° cos 24° + sin 84° sin 24° using your calculator. Did you get a value of 0.5?

Here’s why: cos 84° cos 24° + sin 84° sin 24° = cos (84° - 24°) = cos 60° = 0.5

Example 6: Use the identity for the cosine of the difference of two angles to simplify cos (p - x).

cos (p - x) = cos p cos x + sin p sin x = (-1) cos x + 0•sin x = -cos x.

Problems: Use the identity for cos (u - v) to simplify each of the following.

1.

2.

Now refer to your text to see where the identity for the cosine of the difference of two angles comes from and find some more trig identities for the sum and difference of angles. Remember that these identities are the only way to simplify sums and differences because we found that there was NOT a distributive law for applying trig functions to angles; we will have to work hard at recognizing the situation.

Answers:

1. sin x

2. –sin q