I Have Questions. You Have The Answers. (6)

Suppose tan⁡xtan⁡y=13\large \frac{\tan x}{\tan y} = \frac{1}{3} and sin⁡2xsin⁡2y=34,\large \frac{\sin 2x}{\sin 2y} = \frac{3}{4}, where 0<x,y<π20 < x, y < \frac{\pi}{2}. What is the value of tan⁡2xtan⁡2y?\large \frac{\tan 2x}{\tan 2y}?

Note by Benedict Dimacutac
3 years, 7 months ago

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Comments

Hint: Expand the first equation by expressing tan=sin/cos

Hint 2: Expand the second equation by using the identity sin(2A) = 2sin A cos A

Hint 3: Now you got 2 equations in terms of sin and cos only. Take their ratio. Show that 4cos^2 x = 9 cos^2 y

Hint 4: Let Z = tan(2x) / tan(2y), find Z^2 instead. Apply tan^2 (2A) + 1 = sec^2(2A), sec^2(2A) = 1/cos^2(2A), cos(2A) = 2cos(A) + 1

HInt 5: Show that Z<0. And get Z = -3/11

Pi Han Goh - 3 years, 4 months ago
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