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Sagot :
To find the exact values of [tex]\(\sec \theta\)[/tex] and [tex]\(\cot \theta\)[/tex] given that [tex]\(\sin \theta = -\frac{4}{9}\)[/tex] and [tex]\(\theta\)[/tex] is in quadrant IV, follow these steps:
1. Understand the given information:
- [tex]\(\sin \theta = -\frac{4}{9}\)[/tex]
- [tex]\(\theta\)[/tex] is in quadrant IV, where sine is negative and cosine is positive.
2. Use the Pythagorean identity to find [tex]\(\cos \theta\)[/tex]:
The Pythagorean identity states that:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
Substitute [tex]\(\sin \theta = -\frac{4}{9}\)[/tex]:
[tex]\[ \left(-\frac{4}{9}\right)^2 + \cos^2 \theta = 1 \][/tex]
Simplify:
[tex]\[ \frac{16}{81} + \cos^2 \theta = 1 \][/tex]
Solve for [tex]\(\cos^2 \theta\)[/tex]:
[tex]\[ \cos^2 \theta = 1 - \frac{16}{81} \][/tex]
[tex]\[ \cos^2 \theta = \frac{81}{81} - \frac{16}{81} \][/tex]
[tex]\[ \cos^2 \theta = \frac{65}{81} \][/tex]
Therefore:
[tex]\[ \cos \theta = \sqrt{\frac{65}{81}} \][/tex]
Since [tex]\(\theta\)[/tex] is in quadrant IV where cosine is positive:
[tex]\[ \cos \theta = \frac{\sqrt{65}}{9} \][/tex]
3. Calculate [tex]\(\sec \theta\)[/tex]:
By definition,
[tex]\[ \sec \theta = \frac{1}{\cos \theta} \][/tex]
Substituting [tex]\(\cos \theta = \frac{\sqrt{65}}{9}\)[/tex]:
[tex]\[ \sec \theta = \frac{1}{\frac{\sqrt{65}}{9}} = \frac{9}{\sqrt{65}} \][/tex]
Rationalize the denominator:
[tex]\[ \sec \theta = \frac{9\sqrt{65}}{65} \][/tex]
4. Calculate [tex]\(\cot \theta\)[/tex]:
Recall that:
[tex]\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \][/tex]
Substitute [tex]\(\cos \theta = \frac{\sqrt{65}}{9}\)[/tex] and [tex]\(\sin \theta = -\frac{4}{9}\)[/tex]:
[tex]\[ \cot \theta = \frac{\frac{\sqrt{65}}{9}}{-\frac{4}{9}} = \frac{\sqrt{65}}{-4} = -\frac{\sqrt{65}}{4} \][/tex]
So, the exact values are:
[tex]\[ \sec \theta = \frac{9\sqrt{65}}{65} \][/tex]
[tex]\[ \cot \theta = -\frac{\sqrt{65}}{4} \][/tex]
Given the numerical answers:
[tex]\[ \sec \theta \approx 1.116312611302876 \][/tex]
[tex]\[ \cot \theta \approx -2.0155644370746373 \][/tex]
1. Understand the given information:
- [tex]\(\sin \theta = -\frac{4}{9}\)[/tex]
- [tex]\(\theta\)[/tex] is in quadrant IV, where sine is negative and cosine is positive.
2. Use the Pythagorean identity to find [tex]\(\cos \theta\)[/tex]:
The Pythagorean identity states that:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
Substitute [tex]\(\sin \theta = -\frac{4}{9}\)[/tex]:
[tex]\[ \left(-\frac{4}{9}\right)^2 + \cos^2 \theta = 1 \][/tex]
Simplify:
[tex]\[ \frac{16}{81} + \cos^2 \theta = 1 \][/tex]
Solve for [tex]\(\cos^2 \theta\)[/tex]:
[tex]\[ \cos^2 \theta = 1 - \frac{16}{81} \][/tex]
[tex]\[ \cos^2 \theta = \frac{81}{81} - \frac{16}{81} \][/tex]
[tex]\[ \cos^2 \theta = \frac{65}{81} \][/tex]
Therefore:
[tex]\[ \cos \theta = \sqrt{\frac{65}{81}} \][/tex]
Since [tex]\(\theta\)[/tex] is in quadrant IV where cosine is positive:
[tex]\[ \cos \theta = \frac{\sqrt{65}}{9} \][/tex]
3. Calculate [tex]\(\sec \theta\)[/tex]:
By definition,
[tex]\[ \sec \theta = \frac{1}{\cos \theta} \][/tex]
Substituting [tex]\(\cos \theta = \frac{\sqrt{65}}{9}\)[/tex]:
[tex]\[ \sec \theta = \frac{1}{\frac{\sqrt{65}}{9}} = \frac{9}{\sqrt{65}} \][/tex]
Rationalize the denominator:
[tex]\[ \sec \theta = \frac{9\sqrt{65}}{65} \][/tex]
4. Calculate [tex]\(\cot \theta\)[/tex]:
Recall that:
[tex]\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \][/tex]
Substitute [tex]\(\cos \theta = \frac{\sqrt{65}}{9}\)[/tex] and [tex]\(\sin \theta = -\frac{4}{9}\)[/tex]:
[tex]\[ \cot \theta = \frac{\frac{\sqrt{65}}{9}}{-\frac{4}{9}} = \frac{\sqrt{65}}{-4} = -\frac{\sqrt{65}}{4} \][/tex]
So, the exact values are:
[tex]\[ \sec \theta = \frac{9\sqrt{65}}{65} \][/tex]
[tex]\[ \cot \theta = -\frac{\sqrt{65}}{4} \][/tex]
Given the numerical answers:
[tex]\[ \sec \theta \approx 1.116312611302876 \][/tex]
[tex]\[ \cot \theta \approx -2.0155644370746373 \][/tex]
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