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 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

leaving

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.

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.