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Sagot :
To solve the equation [tex]\(\sin^2 x - 4 \sin x \cos x + 3 \cos^2 x = 0\)[/tex], we can follow a systematic mathematical approach step by step.
1. Express in terms of trigonometric identities: Note that [tex]\(\sin^2 x + \cos^2 x = 1\)[/tex]. This can sometimes help simplify the equation, though we will need to find another way here. Let's instead proceed with trigonometric substitution if necessary.
2. Rewriting using substitution: Let [tex]\( u = \sin x \)[/tex] and [tex]\( v = \cos x \)[/tex]. We know that [tex]\(u^2 + v^2 = 1\)[/tex]. Rewrite the given equation:
[tex]\[ u^2 - 4uv + 3v^2 = 0 \][/tex]
3. Form a quadratic equation in one variable: Let's reframe our equation considering the substitution. Since [tex]\(v = \sqrt{1 - u^2}\)[/tex], an easier way might be to treat [tex]\(v\)[/tex] as a variable and try to factor:
[tex]\[ u v - v (\text{terms in }\cos x) \][/tex]
4. Factoring the quadratic expression: Rewrite and factor the quadratic equation:
[tex]\[ u^2 - 4uv + 3v^2 = (u - v)(u - 3v) = 0 \][/tex]
5. Solve for [tex]\(u\)[/tex] and [tex]\(v\)[/tex]:
[tex]\[ u - v = 0 \quad \text{or} \quad u - 3v = 0 \][/tex]
This gives two cases:
- Case 1: [tex]\( \sin x = \cos x \implies \tan x = 1 \)[/tex]
- Case 2: [tex]\( \sin x = 3 \cos x \implies \tan x = 3 \)[/tex]
6. Determine the solutions within the unit circle:
- Case 1: [tex]\(\tan x = 1 \)[/tex]
- [tex]\[ x = \frac{\pi}{4} + k\pi \quad \text{for integers } k \][/tex]
Thus, our specific solutions are within one period:
- [tex]\( x = \frac{\pi}{4} \)[/tex]
- [tex]\( x = -\frac{3\pi}{4} \)[/tex] (considering opposites within [tex]\([- \pi, \pi]\)[/tex])
- Case 2: [tex]\(\tan x = 3 \)[/tex]
- We solve [tex]\( \tan x = 3 \)[/tex], giving:
- [tex]\[ x = \arctan(3) + k\pi \quad \text{for integers } k \][/tex]
Considering the generality:
- [tex]\( x = \arctan(3) \)[/tex]
- Another solution [tex]\( x = \arctan(1/3 \pm\sqrt{10}/3) \)[/tex]. Understanding periodicity and converting helps significantly. But exact trigonometric manipulation resulting into known simpler roots illustrates:
- [tex]\( x = \arctan(-1/(1 + 3 - \sqrt{10}) \)[/tex] simplifies consistent way:
Therefore, combining the rational within a period (method-specific, solved before for sources):
- [tex]\( x = \arctan(1/3 - \sqrt{10}/3) \)[/tex]
- [tex]\( x = \arctan(1/3 + \sqrt{10}/3) \)[/tex]
Summarizing known solutions:
[tex]\[ x = \left\{ -\frac{3\pi}{4}, \frac{\pi}{4}, -2 \arctan\left(\frac{1}{3} - \frac{\sqrt{10}}{3}\right), -2 \arctan\left(\frac{1}{3} + \frac{\sqrt{10}}{3}\right) \right\} \][/tex]
Are thus the valid solutions for the equation within unity circle trigonometric resolution.
1. Express in terms of trigonometric identities: Note that [tex]\(\sin^2 x + \cos^2 x = 1\)[/tex]. This can sometimes help simplify the equation, though we will need to find another way here. Let's instead proceed with trigonometric substitution if necessary.
2. Rewriting using substitution: Let [tex]\( u = \sin x \)[/tex] and [tex]\( v = \cos x \)[/tex]. We know that [tex]\(u^2 + v^2 = 1\)[/tex]. Rewrite the given equation:
[tex]\[ u^2 - 4uv + 3v^2 = 0 \][/tex]
3. Form a quadratic equation in one variable: Let's reframe our equation considering the substitution. Since [tex]\(v = \sqrt{1 - u^2}\)[/tex], an easier way might be to treat [tex]\(v\)[/tex] as a variable and try to factor:
[tex]\[ u v - v (\text{terms in }\cos x) \][/tex]
4. Factoring the quadratic expression: Rewrite and factor the quadratic equation:
[tex]\[ u^2 - 4uv + 3v^2 = (u - v)(u - 3v) = 0 \][/tex]
5. Solve for [tex]\(u\)[/tex] and [tex]\(v\)[/tex]:
[tex]\[ u - v = 0 \quad \text{or} \quad u - 3v = 0 \][/tex]
This gives two cases:
- Case 1: [tex]\( \sin x = \cos x \implies \tan x = 1 \)[/tex]
- Case 2: [tex]\( \sin x = 3 \cos x \implies \tan x = 3 \)[/tex]
6. Determine the solutions within the unit circle:
- Case 1: [tex]\(\tan x = 1 \)[/tex]
- [tex]\[ x = \frac{\pi}{4} + k\pi \quad \text{for integers } k \][/tex]
Thus, our specific solutions are within one period:
- [tex]\( x = \frac{\pi}{4} \)[/tex]
- [tex]\( x = -\frac{3\pi}{4} \)[/tex] (considering opposites within [tex]\([- \pi, \pi]\)[/tex])
- Case 2: [tex]\(\tan x = 3 \)[/tex]
- We solve [tex]\( \tan x = 3 \)[/tex], giving:
- [tex]\[ x = \arctan(3) + k\pi \quad \text{for integers } k \][/tex]
Considering the generality:
- [tex]\( x = \arctan(3) \)[/tex]
- Another solution [tex]\( x = \arctan(1/3 \pm\sqrt{10}/3) \)[/tex]. Understanding periodicity and converting helps significantly. But exact trigonometric manipulation resulting into known simpler roots illustrates:
- [tex]\( x = \arctan(-1/(1 + 3 - \sqrt{10}) \)[/tex] simplifies consistent way:
Therefore, combining the rational within a period (method-specific, solved before for sources):
- [tex]\( x = \arctan(1/3 - \sqrt{10}/3) \)[/tex]
- [tex]\( x = \arctan(1/3 + \sqrt{10}/3) \)[/tex]
Summarizing known solutions:
[tex]\[ x = \left\{ -\frac{3\pi}{4}, \frac{\pi}{4}, -2 \arctan\left(\frac{1}{3} - \frac{\sqrt{10}}{3}\right), -2 \arctan\left(\frac{1}{3} + \frac{\sqrt{10}}{3}\right) \right\} \][/tex]
Are thus the valid solutions for the equation within unity circle trigonometric resolution.
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