Solution to Exercise 7.1-1

The problem was to come up with an identity for sin³(x).

Step 1: Break out what you already have an identity for. As mentioned in the problem, we already have an identity that

   sin²(x)  =  0.5(1 - cos(2x) )

And you know that sin²(x) is a factor of sin³(x). So you have

   sin³(x)  =  sin(x)sin²(x)  =  0.5 sin(x) (1 - cos(2x) )

Step 2: Multiply it out. This gives you

   sin³(x) = 0.5 sin(x) - 0.5 sin(x)cos(2x)

which is already interesting, but you can take it further.

Step 3: Recall the identity for sin(2x + x). That is, simply apply the identity for the sine of the sum of two angles to the sum of 2x and x.

   sin(3x)  =  sin(2x + x)  =  sin(2x)cos(x) + sin(x)cos(2x)

The rightmost summand looks familiar doesn't it? Indeed it appears in step 2. But what do we do with the left-hand summand??

Step 4: Observe that sin(x) is the same as sin(2x - x). And applying the identity for sine of a difference to that you get

   sin(x)  =  sin(2x - x)  =  sin(2x)cos(x) - sin(x)cos(2x)

Step 5: Put steps 3 and 4 together. By subtracting step 4 from step 3, you get a useful intermediate identity

   sin(3x) - sin(x)  =  2 sin(x)cos(2x)

   0.25 ( sin(3x) - sin(x) )  =  0.5 sin(x)cos(2x)

Step 6: Substitute step 5 into step 2. This gives you

   sin³(x)  =  0.5 sin(x) - 0.25 ( sin(3x) - sin(x) )

or, gathering like terms

   sin³(x)  =  0.75 sin(x) - 0.25 sin(3x)

which is the identity the problem was after.

Comments: We discussed earlier that sin(x) is an odd function, that is sin(-x) = -sin(x). When you raise any odd function to an odd power, the result is another odd function (analogous to the fact that when you multiply an odd number by an odd number, the result is odd). In this case, we cubed sin(x) and got the sum of two sines, each of which is an odd function. And the sum of any two odd functions is also odd.

We also discussed that cos(x) is an even function, that is cos(-x) = cos(x). Notice that when we squared sin(x) we got 0.5(1 - cos(2x)), which is an even function. Whenever you raise an odd function to an even power, the result is an even function (again the analogy to multiplying an odd number by an even number results in an even number).

The point here is that if you can determine by inspection whether an answer you come up with is an even or an odd function, you can often use that fact to do a cursory check of your work. In the case of the identity for sin³(x), we would be expecting the result to be an odd function. If by inspecting you found your result not to be odd, then you would know you had made a mistake.

What do you think the result is of raising an even function to an even power or an odd power?

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