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[Solved] Trigonometric equations

 
Trigonometry 1 & 2
Last Post by Staff 4 years ago
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June 15, 2021 2:21 pm
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(@armor)
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1623763289-fG2TXMSnBIyyhIKl5oJdTqpGbSAayXCL-img_8262.jpg

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June 15, 2021 2:21 pm
Topic starter
Staff 
(@armor)
Admin
  Honorable Member
317 Posts
156 153 0

\small tan^{-1}x+cot^{-1}y=tan^{-1}3

\small \Rightarrow tan^{-1}x+tan^{-1}\frac{1}{y}=tan^{-1}3\Rightarrow tan(tan^{-1}x+tan^{-1\frac{1}{y}})=tan(tan^{-1}3)

\small \Rightarrow \frac{x+\frac{1}{y}}{1-\frac{x}{y}}=3\Rightarrow x=\frac{3y-1}{y+3}=3-\frac{10}{y+3} .

Since x, y are positive integers, then y+3 is the divisor of 10. Thus y=2 or y=7,  x=1 or x=2. As a conclusion, the positive integer solutions are x=1 y=2 or x=2 y=7.

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