Answer :
To find [tex]\(\cos(\theta)\)[/tex] given that [tex]\(\sin(\theta) = \frac{7}{12}\)[/tex], we can use the Pythagorean identity:
[tex]\[ \sin^2(\theta) + \cos^2(\theta) = 1. \][/tex]
1. Calculate [tex]\(\sin^2(\theta)\)[/tex]:
[tex]\[ \sin^2(\theta) = \left(\frac{7}{12}\right)^2 = \frac{49}{144}. \][/tex]
2. Substitute [tex]\(\sin^2(\theta)\)[/tex] into the Pythagorean identity to find [tex]\(\cos^2(\theta)\)[/tex]:
[tex]\[ \cos^2(\theta) = 1 - \sin^2(\theta) = 1 - \frac{49}{144}. \][/tex]
3. Convert the whole number 1 to a fraction with the same denominator:
[tex]\[ 1 = \frac{144}{144}. \][/tex]
4. Subtract the fractions:
[tex]\[ \cos^2(\theta) = \frac{144}{144} - \frac{49}{144} = \frac{95}{144}. \][/tex]
5. Take the square root of both sides to find [tex]\(\cos(\theta)\)[/tex]:
[tex]\[ \cos(\theta) = \sqrt{\frac{95}{144}} = \frac{\sqrt{95}}{12}. \][/tex]
Therefore, [tex]\(\cos(\theta)\)[/tex] is:
[tex]\[ \boxed{\frac{\sqrt{95}}{12}}. \][/tex]
So, the correct answer is:
[tex]\[ A. \frac{\sqrt{95}}{12}. \][/tex]
[tex]\[ \sin^2(\theta) + \cos^2(\theta) = 1. \][/tex]
1. Calculate [tex]\(\sin^2(\theta)\)[/tex]:
[tex]\[ \sin^2(\theta) = \left(\frac{7}{12}\right)^2 = \frac{49}{144}. \][/tex]
2. Substitute [tex]\(\sin^2(\theta)\)[/tex] into the Pythagorean identity to find [tex]\(\cos^2(\theta)\)[/tex]:
[tex]\[ \cos^2(\theta) = 1 - \sin^2(\theta) = 1 - \frac{49}{144}. \][/tex]
3. Convert the whole number 1 to a fraction with the same denominator:
[tex]\[ 1 = \frac{144}{144}. \][/tex]
4. Subtract the fractions:
[tex]\[ \cos^2(\theta) = \frac{144}{144} - \frac{49}{144} = \frac{95}{144}. \][/tex]
5. Take the square root of both sides to find [tex]\(\cos(\theta)\)[/tex]:
[tex]\[ \cos(\theta) = \sqrt{\frac{95}{144}} = \frac{\sqrt{95}}{12}. \][/tex]
Therefore, [tex]\(\cos(\theta)\)[/tex] is:
[tex]\[ \boxed{\frac{\sqrt{95}}{12}}. \][/tex]
So, the correct answer is:
[tex]\[ A. \frac{\sqrt{95}}{12}. \][/tex]
