How solve this trigo system ?
01-31-2020, 10:48 PM
Post: #12
 Albert Chan Senior Member Posts: 2,355 Joined: Jul 2018
RE: How solve this trigo system ?
(09-18-2017 08:56 AM)ggauny@live.fr Wrote:  cos x - cos y = 1/2
sin x * sin y = 3/8

Nice problem !
Square both side, both equations:

$$\cos^2 x - 2 \cos x \cos y + \cos^2 y = 1/4$$
$$(1-\cos^2 x)(1-\cos^2 y) = 1 - (\cos^2 x + \cos^2 y) + (\cos^2 x \cos^2 y) = 9/64$$

$$(\cos x \cos y)^2 - 2( \cos x \cos y) + 39/64 = 0 \quad\; → \cos x \cos y = 3/8$$

Using angle sum formula, we have:

$$\cos(x + y) = \cos x \cos y - \sin x \sin y = 0 \quad\quad → x + y = ±\,90°$$

This meant we can replace sin(y) as ±cos(x)

2 sin(x) cos(x) = sin(2x) = ±3/4

The process of squaring may introduce some bad solutions.
With those eliminated, we have x ≈ ±24.295°, ±114.295°
These solutions corresponded to y ≈ ±65.705°, ±155.705°
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 Messages In This Thread How solve this trigo system ? - ggauny@live.fr - 09-18-2017, 08:56 AM RE: How solve this trigo system ? - Gerson W. Barbosa - 09-18-2017, 10:12 AM RE: How solve this trigo system ? - ggauny@live.fr - 09-18-2017, 10:45 AM RE: How solve this trigo system ? - Gerson W. Barbosa - 09-18-2017, 10:55 AM RE: How solve this trigo system ? - Gerson W. Barbosa - 09-18-2017, 11:28 AM RE: How solve this trigo system ? - Dieter - 09-18-2017, 12:46 PM RE: How solve this trigo system ? - Dieter - 09-18-2017, 05:48 PM RE: How solve this trigo system ? - AlexFekken - 09-19-2017, 06:36 AM RE: How solve this trigo system ? - Albert Chan - 01-31-2020 10:48 PM RE: How solve this trigo system ? - Dieter - 09-19-2017, 06:30 PM RE: How solve this trigo system ? - ggauny@live.fr - 09-22-2017, 07:27 AM RE: How solve this trigo system ? - Gerson W. Barbosa - 09-22-2017, 05:05 PM

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