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Sagot :
To find the value of [tex]\( n \)[/tex] to the nearest whole number, we will apply the Law of Cosines, which states:
[tex]\[ a^2 = b^2 + c^2 - 2 b c \cos(A) \][/tex]
Given the values:
- [tex]\( a = 22 \)[/tex]
- [tex]\( b = 29 \)[/tex]
- [tex]\( c = 41 \)[/tex]
- [tex]\( A = 18^\circ \)[/tex]
First, we need to convert the angle [tex]\( A \)[/tex] from degrees to radians, because the cosine function typically works with radians in mathematical calculations. The conversion from degrees to radians is done using the formula:
[tex]\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \][/tex]
Therefore, [tex]\( A = 18^\circ \)[/tex] converted to radians is:
[tex]\[ A = 18 \times \frac{\pi}{180} = \frac{\pi}{10} \approx 0.314 \text{ radians} \][/tex]
Applying the Law of Cosines:
[tex]\[ a^2 = b^2 + c^2 - 2 b c \cos(A) \][/tex]
Plugging in the given values:
[tex]\[ 22^2 = 29^2 + 41^2 - 2 \cdot 29 \cdot 41 \cos(0.314) \][/tex]
Calculating the squares:
[tex]\[ 484 = 841 + 1681 - 2 \cdot 29 \cdot 41 \cos(0.314) \][/tex]
Adding the squares:
[tex]\[ 484 = 2522 - 2 \cdot 29 \cdot 41 \cos(0.314) \][/tex]
Now, compute the cosine part. Let's call this value [tex]\( x \)[/tex]:
[tex]\[ x = 2 \cdot 29 \cdot 41 \cos(0.314) \][/tex]
[tex]\[ x \approx 2 \cdot 29 \cdot 41 \cdot 0.951 \approx 2263.262 \][/tex]
Continuing with the equation:
[tex]\[ 484 = 2522 - 2263.262 \][/tex]
Solving this step-by-step:
[tex]\[ 484 = 258.738 \][/tex]
To find the value of [tex]\( n \)[/tex], we solve for [tex]\( n \)[/tex]:
[tex]\[ n^2 = 258.738 \][/tex]
Taking the square root of both sides:
[tex]\[ n = \sqrt{258.738} \approx 16.09 \][/tex]
Therefore, rounding [tex]\( n \)[/tex] to the nearest whole number gives:
[tex]\[ n \approx 16 \][/tex]
Thus, the value of [tex]\( n \)[/tex] to the nearest whole number is:
[tex]\[ \boxed{16} \][/tex]
[tex]\[ a^2 = b^2 + c^2 - 2 b c \cos(A) \][/tex]
Given the values:
- [tex]\( a = 22 \)[/tex]
- [tex]\( b = 29 \)[/tex]
- [tex]\( c = 41 \)[/tex]
- [tex]\( A = 18^\circ \)[/tex]
First, we need to convert the angle [tex]\( A \)[/tex] from degrees to radians, because the cosine function typically works with radians in mathematical calculations. The conversion from degrees to radians is done using the formula:
[tex]\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \][/tex]
Therefore, [tex]\( A = 18^\circ \)[/tex] converted to radians is:
[tex]\[ A = 18 \times \frac{\pi}{180} = \frac{\pi}{10} \approx 0.314 \text{ radians} \][/tex]
Applying the Law of Cosines:
[tex]\[ a^2 = b^2 + c^2 - 2 b c \cos(A) \][/tex]
Plugging in the given values:
[tex]\[ 22^2 = 29^2 + 41^2 - 2 \cdot 29 \cdot 41 \cos(0.314) \][/tex]
Calculating the squares:
[tex]\[ 484 = 841 + 1681 - 2 \cdot 29 \cdot 41 \cos(0.314) \][/tex]
Adding the squares:
[tex]\[ 484 = 2522 - 2 \cdot 29 \cdot 41 \cos(0.314) \][/tex]
Now, compute the cosine part. Let's call this value [tex]\( x \)[/tex]:
[tex]\[ x = 2 \cdot 29 \cdot 41 \cos(0.314) \][/tex]
[tex]\[ x \approx 2 \cdot 29 \cdot 41 \cdot 0.951 \approx 2263.262 \][/tex]
Continuing with the equation:
[tex]\[ 484 = 2522 - 2263.262 \][/tex]
Solving this step-by-step:
[tex]\[ 484 = 258.738 \][/tex]
To find the value of [tex]\( n \)[/tex], we solve for [tex]\( n \)[/tex]:
[tex]\[ n^2 = 258.738 \][/tex]
Taking the square root of both sides:
[tex]\[ n = \sqrt{258.738} \approx 16.09 \][/tex]
Therefore, rounding [tex]\( n \)[/tex] to the nearest whole number gives:
[tex]\[ n \approx 16 \][/tex]
Thus, the value of [tex]\( n \)[/tex] to the nearest whole number is:
[tex]\[ \boxed{16} \][/tex]
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