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Sagot :
To solve the equation [tex]\(\sec(2\theta) = 3\)[/tex], we’ll proceed step-by-step using trigonometric identities to find the value of [tex]\(\cos(\theta)\)[/tex].
1. Express [tex]\(\sec(2\theta)\)[/tex] in terms of [tex]\(\cos(2\theta)\)[/tex]:
Recall that [tex]\(\sec(x) = \frac{1}{\cos(x)}\)[/tex]. Therefore, we can rewrite the given equation:
[tex]\[ \sec(2\theta) = 3 \implies \frac{1}{\cos(2\theta)} = 3 \][/tex]
2. Solve for [tex]\(\cos(2\theta)\)[/tex]:
Invert both sides of the equation to solve for [tex]\(\cos(2\theta)\)[/tex]:
[tex]\[ \cos(2\theta) = \frac{1}{3} \][/tex]
3. Use the double-angle identity for cosine:
The double-angle identity for cosine is [tex]\(\cos(2\theta) = 2\cos^2(\theta) - 1\)[/tex]. Substitute [tex]\(\frac{1}{3}\)[/tex] for [tex]\(\cos(2\theta)\)[/tex]:
[tex]\[ \frac{1}{3} = 2\cos^2(\theta) - 1 \][/tex]
4. Solve the equation for [tex]\(\cos^2(\theta)\)[/tex]:
Isolate [tex]\(\cos^2(\theta)\)[/tex] by first adding 1 to both sides:
[tex]\[ \frac{1}{3} + 1 = 2\cos^2(\theta) \][/tex]
Combine the fractions:
[tex]\[ \frac{1}{3} + \frac{3}{3} = \frac{4}{3} \][/tex]
So, we have:
[tex]\[ \frac{4}{3} = 2\cos^2(\theta) \][/tex]
5. Isolate [tex]\(\cos^2(\theta)\)[/tex]:
Divide both sides by 2:
[tex]\[ \cos^2(\theta) = \frac{\frac{4}{3}}{2} = \frac{4}{3} \times \frac{1}{2} = \frac{2}{3} \][/tex]
6. Solve for [tex]\(\cos(\theta)\)[/tex]:
Take the square root of both sides to find [tex]\(\cos(\theta)\)[/tex]. Remember to include both the positive and negative roots:
[tex]\[ \cos(\theta) = \pm\sqrt{\frac{2}{3}} \][/tex]
7. Simplify the square root:
[tex]\[ \cos(\theta) = \pm \sqrt{\frac{2}{3}} = \pm \frac{\sqrt{2}}{\sqrt{3}} = \pm \frac{\sqrt{2}}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \pm \frac{\sqrt{6}}{3} \][/tex]
Therefore, the solutions are:
- [tex]\(\cos(2\theta) = \frac{1}{3} \)[/tex]
- [tex]\(\cos(\theta) = \sqrt{\frac{2}{3}} \approx 0.816 \)[/tex]
- [tex]\(\cos(\theta) = -\sqrt{\frac{2}{3}} \approx -0.816 \)[/tex]
So the final values are:
- [tex]\(\cos(2\theta) = 0.333\)[/tex]
- [tex]\(\cos(\theta) \approx 0.816\)[/tex]
- [tex]\(\cos(\theta) \approx -0.816\)[/tex]
1. Express [tex]\(\sec(2\theta)\)[/tex] in terms of [tex]\(\cos(2\theta)\)[/tex]:
Recall that [tex]\(\sec(x) = \frac{1}{\cos(x)}\)[/tex]. Therefore, we can rewrite the given equation:
[tex]\[ \sec(2\theta) = 3 \implies \frac{1}{\cos(2\theta)} = 3 \][/tex]
2. Solve for [tex]\(\cos(2\theta)\)[/tex]:
Invert both sides of the equation to solve for [tex]\(\cos(2\theta)\)[/tex]:
[tex]\[ \cos(2\theta) = \frac{1}{3} \][/tex]
3. Use the double-angle identity for cosine:
The double-angle identity for cosine is [tex]\(\cos(2\theta) = 2\cos^2(\theta) - 1\)[/tex]. Substitute [tex]\(\frac{1}{3}\)[/tex] for [tex]\(\cos(2\theta)\)[/tex]:
[tex]\[ \frac{1}{3} = 2\cos^2(\theta) - 1 \][/tex]
4. Solve the equation for [tex]\(\cos^2(\theta)\)[/tex]:
Isolate [tex]\(\cos^2(\theta)\)[/tex] by first adding 1 to both sides:
[tex]\[ \frac{1}{3} + 1 = 2\cos^2(\theta) \][/tex]
Combine the fractions:
[tex]\[ \frac{1}{3} + \frac{3}{3} = \frac{4}{3} \][/tex]
So, we have:
[tex]\[ \frac{4}{3} = 2\cos^2(\theta) \][/tex]
5. Isolate [tex]\(\cos^2(\theta)\)[/tex]:
Divide both sides by 2:
[tex]\[ \cos^2(\theta) = \frac{\frac{4}{3}}{2} = \frac{4}{3} \times \frac{1}{2} = \frac{2}{3} \][/tex]
6. Solve for [tex]\(\cos(\theta)\)[/tex]:
Take the square root of both sides to find [tex]\(\cos(\theta)\)[/tex]. Remember to include both the positive and negative roots:
[tex]\[ \cos(\theta) = \pm\sqrt{\frac{2}{3}} \][/tex]
7. Simplify the square root:
[tex]\[ \cos(\theta) = \pm \sqrt{\frac{2}{3}} = \pm \frac{\sqrt{2}}{\sqrt{3}} = \pm \frac{\sqrt{2}}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \pm \frac{\sqrt{6}}{3} \][/tex]
Therefore, the solutions are:
- [tex]\(\cos(2\theta) = \frac{1}{3} \)[/tex]
- [tex]\(\cos(\theta) = \sqrt{\frac{2}{3}} \approx 0.816 \)[/tex]
- [tex]\(\cos(\theta) = -\sqrt{\frac{2}{3}} \approx -0.816 \)[/tex]
So the final values are:
- [tex]\(\cos(2\theta) = 0.333\)[/tex]
- [tex]\(\cos(\theta) \approx 0.816\)[/tex]
- [tex]\(\cos(\theta) \approx -0.816\)[/tex]
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