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Sagot :
Given that [tex]\(\sin(\theta) = \frac{3 \sqrt{13}}{13}\)[/tex], we need to find the values of the other five trigonometric functions: [tex]\(\cos(\theta)\)[/tex], [tex]\(\tan(\theta)\)[/tex], [tex]\(\cot(\theta)\)[/tex], [tex]\(\sec(\theta)\)[/tex], and [tex]\(\csc(\theta)\)[/tex].
Step 1: Calculate [tex]\(\cos(\theta)\)[/tex]
Use the Pythagorean identity:
[tex]\[ \sin^2(\theta) + \cos^2(\theta) = 1 \][/tex]
Given [tex]\(\sin(\theta) = \frac{3 \sqrt{13}}{13}\)[/tex], calculate [tex]\(\sin^2(\theta)\)[/tex]:
[tex]\[ \sin^2(\theta) = \left( \frac{3 \sqrt{13}}{13} \right)^2 = \frac{9 \times 13}{169} = \frac{117}{169} \][/tex]
Now, solve for [tex]\(\cos^2(\theta)\)[/tex]:
[tex]\[ \cos^2(\theta) = 1 - \sin^2(\theta) = 1 - \frac{117}{169} = \frac{169}{169} - \frac{117}{169} = \frac{52}{169} \][/tex]
Thus, [tex]\(\cos(\theta)\)[/tex] is:
[tex]\[ \cos(\theta) = \sqrt{\frac{52}{169}} = \frac{\sqrt{52}}{13} = \frac{2 \sqrt{13}}{13} \][/tex]
So, [tex]\(\cos(\theta) \approx 0.5547001962252293\)[/tex].
Step 2: Calculate [tex]\(\tan(\theta)\)[/tex]
Use the definition of tangent:
[tex]\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \][/tex]
[tex]\[ \tan(\theta) = \frac{\frac{3 \sqrt{13}}{13}}{\frac{2 \sqrt{13}}{13}} = \frac{3 \sqrt{13}}{13} \times \frac{13}{2 \sqrt{13}} = \frac{3}{2} = 1.5 \][/tex]
So, [tex]\(\tan(\theta) \approx 1.5\)[/tex].
Step 3: Calculate [tex]\(\cot(\theta)\)[/tex]
Use the definition of cotangent:
[tex]\[ \cot(\theta) = \frac{1}{\tan(\theta)} \][/tex]
[tex]\[ \cot(\theta) = \frac{1}{1.5} = \frac{2}{3} \approx 0.6666666666666666 \][/tex]
So, [tex]\(\cot(\theta) \approx 0.6666666666666669\)[/tex].
Step 4: Calculate [tex]\(\sec(\theta)\)[/tex]
Use the definition of secant:
[tex]\[ \sec(\theta) = \frac{1}{\cos(\theta)} \][/tex]
[tex]\[ \sec(\theta) = \frac{1}{\frac{2 \sqrt{13}}{13}} = \frac{13}{2 \sqrt{13}} = \frac{13}{2 \sqrt{13}} \times \frac{\sqrt{13}}{\sqrt{13}} = \frac{13 \sqrt{13}}{26} = \frac{\sqrt{13}}{2} \approx 1.8027756377319946 \][/tex]
So, [tex]\(\sec(\theta) \approx 1.8027756377319941\)[/tex].
Step 5: Calculate [tex]\(\csc(\theta)\)[/tex]
Use the definition of cosecant:
[tex]\[ \csc(\theta) = \frac{1}{\sin(\theta)} \][/tex]
[tex]\[ \csc(\theta) = \frac{1}{\frac{3 \sqrt{13}}{13}} = \frac{13}{3 \sqrt{13}} = \frac{13}{3 \sqrt{13}} \times \frac{\sqrt{13}}{\sqrt{13}} = \frac{13 \sqrt{13}}{39} = \frac{\sqrt{13}}{3} \approx 1.2018504251546645 \][/tex]
So, [tex]\(\csc(\theta) \approx 1.2018504251546631\)[/tex].
In summary, the functions are as follows:
- [tex]\(\sin(\theta) \approx 0.8320502943378436\)[/tex]
- [tex]\(\cos(\theta) \approx 0.5547001962252293\)[/tex]
- [tex]\(\tan(\theta) \approx 1.5\)[/tex]
- [tex]\(\cot(\theta) \approx 0.6666666666666669\)[/tex]
- [tex]\(\sec(\theta) \approx 1.8027756377319941\)[/tex]
- [tex]\(\csc(\theta) \approx 1.2018504251546631\)[/tex]
Step 1: Calculate [tex]\(\cos(\theta)\)[/tex]
Use the Pythagorean identity:
[tex]\[ \sin^2(\theta) + \cos^2(\theta) = 1 \][/tex]
Given [tex]\(\sin(\theta) = \frac{3 \sqrt{13}}{13}\)[/tex], calculate [tex]\(\sin^2(\theta)\)[/tex]:
[tex]\[ \sin^2(\theta) = \left( \frac{3 \sqrt{13}}{13} \right)^2 = \frac{9 \times 13}{169} = \frac{117}{169} \][/tex]
Now, solve for [tex]\(\cos^2(\theta)\)[/tex]:
[tex]\[ \cos^2(\theta) = 1 - \sin^2(\theta) = 1 - \frac{117}{169} = \frac{169}{169} - \frac{117}{169} = \frac{52}{169} \][/tex]
Thus, [tex]\(\cos(\theta)\)[/tex] is:
[tex]\[ \cos(\theta) = \sqrt{\frac{52}{169}} = \frac{\sqrt{52}}{13} = \frac{2 \sqrt{13}}{13} \][/tex]
So, [tex]\(\cos(\theta) \approx 0.5547001962252293\)[/tex].
Step 2: Calculate [tex]\(\tan(\theta)\)[/tex]
Use the definition of tangent:
[tex]\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \][/tex]
[tex]\[ \tan(\theta) = \frac{\frac{3 \sqrt{13}}{13}}{\frac{2 \sqrt{13}}{13}} = \frac{3 \sqrt{13}}{13} \times \frac{13}{2 \sqrt{13}} = \frac{3}{2} = 1.5 \][/tex]
So, [tex]\(\tan(\theta) \approx 1.5\)[/tex].
Step 3: Calculate [tex]\(\cot(\theta)\)[/tex]
Use the definition of cotangent:
[tex]\[ \cot(\theta) = \frac{1}{\tan(\theta)} \][/tex]
[tex]\[ \cot(\theta) = \frac{1}{1.5} = \frac{2}{3} \approx 0.6666666666666666 \][/tex]
So, [tex]\(\cot(\theta) \approx 0.6666666666666669\)[/tex].
Step 4: Calculate [tex]\(\sec(\theta)\)[/tex]
Use the definition of secant:
[tex]\[ \sec(\theta) = \frac{1}{\cos(\theta)} \][/tex]
[tex]\[ \sec(\theta) = \frac{1}{\frac{2 \sqrt{13}}{13}} = \frac{13}{2 \sqrt{13}} = \frac{13}{2 \sqrt{13}} \times \frac{\sqrt{13}}{\sqrt{13}} = \frac{13 \sqrt{13}}{26} = \frac{\sqrt{13}}{2} \approx 1.8027756377319946 \][/tex]
So, [tex]\(\sec(\theta) \approx 1.8027756377319941\)[/tex].
Step 5: Calculate [tex]\(\csc(\theta)\)[/tex]
Use the definition of cosecant:
[tex]\[ \csc(\theta) = \frac{1}{\sin(\theta)} \][/tex]
[tex]\[ \csc(\theta) = \frac{1}{\frac{3 \sqrt{13}}{13}} = \frac{13}{3 \sqrt{13}} = \frac{13}{3 \sqrt{13}} \times \frac{\sqrt{13}}{\sqrt{13}} = \frac{13 \sqrt{13}}{39} = \frac{\sqrt{13}}{3} \approx 1.2018504251546645 \][/tex]
So, [tex]\(\csc(\theta) \approx 1.2018504251546631\)[/tex].
In summary, the functions are as follows:
- [tex]\(\sin(\theta) \approx 0.8320502943378436\)[/tex]
- [tex]\(\cos(\theta) \approx 0.5547001962252293\)[/tex]
- [tex]\(\tan(\theta) \approx 1.5\)[/tex]
- [tex]\(\cot(\theta) \approx 0.6666666666666669\)[/tex]
- [tex]\(\sec(\theta) \approx 1.8027756377319941\)[/tex]
- [tex]\(\csc(\theta) \approx 1.2018504251546631\)[/tex]
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