Answered

Select the correct answer.

If [tex]\cos (\theta) = \frac{3}{10}[/tex], what is [tex]\sin (\theta)[/tex]?

A. [tex]\frac{7}{10}[/tex]

B. [tex]\frac{\sqrt{91}}{10}[/tex]

C. [tex]\frac{91}{100}[/tex]

D. [tex]\frac{\sqrt{7}}{100}[/tex]



Answer :

GinnyAnswer
To find [tex]\(\sin(\theta)\)[/tex] given that [tex]\(\cos(\theta) = \frac{3}{10}\)[/tex], we can use the Pythagorean identity, which states:

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

1. We are given [tex]\(\cos(\theta) = \frac{3}{10}\)[/tex]. First, we calculate [tex]\(\cos^2(\theta)\)[/tex]:

[tex]\[ \cos^2(\theta) = \left(\frac{3}{10}\right)^2 = \frac{9}{100} \][/tex]

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

[tex]\[ \sin^2(\theta) = 1 - \cos^2(\theta) = 1 - \frac{9}{100} = \frac{100}{100} - \frac{9}{100} = \frac{91}{100} \][/tex]

3. To find [tex]\(\sin(\theta)\)[/tex], we take the positive square root of [tex]\(\sin^2(\theta)\)[/tex] (since [tex]\(\theta\)[/tex] typically represents an angle in the first quadrant where sine is positive):

[tex]\[ \sin(\theta) = \sqrt{\frac{91}{100}} = \frac{\sqrt{91}}{10} \][/tex]

Thus, the value of [tex]\(\sin(\theta)\)[/tex] is:

[tex]\[ \sin(\theta) = \frac{\sqrt{91}}{10} \][/tex]

Therefore, the correct answer is:
B. [tex]\(\frac{\sqrt{91}}{10}\)[/tex]

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