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Further Mathematics : Finding angle of a complex number kinda...

The question had two variables P and Q which were defined as geometric series. Last part of the question asked me to find angle in the range 0 < θ < π for which P + iQ is a real number. Since real numbers have no complex component, that means P + iQ lies on the x - axis, and only possible angles can be 0 or π, which is already outside the range. Can someone explain me please how it is possible that θ is within that range?

Here's the question: https://imgur.com/a/v8sQNrL
Original post by L15032015
The question had two variables P and Q which were defined as geometric series. Last part of the question asked me to find angle in the range 0 < θ < π for which P + iQ is a real number. Since real numbers have no complex component, that means P + iQ lies on the x - axis, and only possible angles can be 0 or π, which is already outside the range. Can someone explain me please how it is possible that θ is within that range?

Here's the question: https://imgur.com/a/v8sQNrL


Not quite, you're confusing yourself.

P+iQP+iQ being real implies that Q=0Q = 0.

So sin⁡θ+sin⁡2θ+…+sin⁡12θ=0\sin \theta + \sin 2 \theta + \ldots + \sin 12\theta = 0.

Sure, θ=0,π\theta = 0, \pi are two possibilities as you say, but they are no in the range, and there are further solutions for which this equation is true which lie within the required interval.
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Original post by RDKGames
Not quite, you're confusing yourself.

P+iQP+iQ being real implies that Q=0Q = 0.

So sin⁡θ+sin⁡2θ+…+sin⁡12θ=0\sin \theta + \sin 2 \theta + \ldots + \sin 12\theta = 0.

Sure, θ=0,π\theta = 0, \pi are two possibilities as you say, but they are no in the range, and there are further solutions for which this equation is true which lie within the required interval.


I tried solving Q = 0, but I thought I was doing it wrong since trigonometric equation that I got seemed odd to me. I have now realized that I haven't practiced enough solving such trig equations. Now at least I know I am on the right track, thank you for your input.

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