Answered

Select the correct answer.

If [tex]\(\sin(\theta) = \frac{7}{12}\)[/tex], what is [tex]\(\cos(\theta)\)[/tex]?

A. [tex]\(\frac{\sqrt{95}}{12}\)[/tex]

B. [tex]\(\frac{5}{12}\)[/tex]

C. [tex]\(\frac{95}{144}\)[/tex]

D. [tex]\(\frac{\sqrt{5}}{12}\)[/tex]



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]