Select the correct answer.

If [tex]\sin (\theta) = \frac{3}{8}[/tex], what is [tex]\cos (\theta)[/tex]?

A. [tex]\frac{5}{8}[/tex]
B. [tex]\frac{\sqrt{55}}{8}[/tex]
C. [tex]\frac{\sqrt{5}}{8}[/tex]
D. [tex]\frac{55}{64}[/tex]



Answer :

To find [tex]\(\cos(\theta)\)[/tex] given that [tex]\(\sin(\theta) = \frac{3}{8}\)[/tex], we can use the Pythagorean identity, which states:

[tex]\[ \sin^2(\theta) + \cos^2(\theta) = 1 \][/tex]

First, we need to compute [tex]\(\sin^2(\theta)\)[/tex]:

[tex]\[ \sin^2(\theta) = \left( \frac{3}{8} \right)^2 = \frac{9}{64} \][/tex]

Using the Pythagorean identity, solve for [tex]\(\cos^2(\theta)\)[/tex]:

[tex]\[ \cos^2(\theta) = 1 - \sin^2(\theta) = 1 - \frac{9}{64} = \frac{64}{64} - \frac{9}{64} = \frac{55}{64} \][/tex]

Next, take the square root of both sides to solve for [tex]\(\cos(\theta)\)[/tex]:

[tex]\[ \cos(\theta) = \sqrt{\frac{55}{64}} \][/tex]

Simplifying the square root:

[tex]\[ \cos(\theta) = \frac{\sqrt{55}}{8} \][/tex]

So the correct answer is:

B. [tex]\(\frac{\sqrt{55}}{8}\)[/tex]