Math CentralQuandaries & Queries


Question from ben, a student:

find all positive angles x such that 3x is one of the nonright angles in a right triangle and sin(2x) = cos(3x).

Hi Ben.

We had fun with this problem, in fact we solved it three ways!

Method 1: First we used the trig identities for sin(2x) and cos(2x), first writing cos(3x) = cos(x + 2x), and arrived at a quadratic in sin(x) which we solved to find sin(x).

Method 2: Then we realized that the subject line of your email was calculus, so although you might want to solve this problem using a formula we learned in first year calculus (it's quicker than using just trig identities anyway). Here's how that works out:

Abraham de Moivre was a French-born mathematician who pioneered the development of analytic geometry and the theory of probability. This is the theory named for him:

cos(nx) + i sin (nx) = (cos x + i sin x)n .

This means that for n=3 we have:

cos(3x) + i sin (3x) = (cos x + i sin x)3
= cos3 x + 3i cos2 x sin x - 3cos x sin2 x - i sin3 x

whereupon we can ignore the imaginary portions and leave the real part:

cos (3x) = cos3 x - 3cos x sin2 x

Using the same reasoning, we can show that sin (2x) = 2 sin x cos x, which is a well-known trig identity anway. Now since sin (2x) = cos (3x), we have:

2 sin x cos x = cos3 x - 3cos x sin2 x
2 sin x = cos2 x - 3sin2 x

and since sin2 x + cos2 x = 1,

2 sin x = (1 - sin2 x) - 3sin2 x


4sin2 x + 2sin x - 1 = 0

which is just a quadratic equation that you can solve with the quadratic formula.

(This is the same quadratic we found using our first method.)

Method 3: Finally we realized that there is another trig identity you might use. It is that

cos(90o - y) = sin(y)

Hence sin(2x) = cos(90o - 2x) and if sin(2x) = cos(3x) you have

cos(90o - 2x) = cos(3x)

So does this mean that 90o - 2x = 3x? In general this is not true because of the periodic nature of cosine, but in your situation the angle 3x is small and positive, 0 < 3x < 90o, and hence in this particular case,

cos(90o - 2x) = cos(3x) does indeed imply that 90o - 2x = 3x.

Whatever method you use, be sure to check your answer(s) to make sure that 3x is a valid angle as described in the original question.

Hope this helps,
Stephen and Penny.


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