Select the correct answer.

Consider this equation:
[tex]\[
\cos (\theta)=-\frac{2 \sqrt{5}}{5}
\][/tex]

If [tex]\(\theta\)[/tex] is an angle in quadrant II, what is the value of [tex]\(\sin (\theta)\)[/tex]?

A. [tex]\(\frac{1}{2}\)[/tex]

B. [tex]\(-\frac{1}{2}\)[/tex]

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

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



Answer :

To find the value of [tex]\(\sin(\theta)\)[/tex] given that [tex]\(\cos(\theta) = -\frac{2\sqrt{5}}{5}\)[/tex] and [tex]\(\theta\)[/tex] is an angle in quadrant II, we can use the Pythagorean identity:

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

Step-by-step solution:

1. Substitute the value of [tex]\(\cos(\theta)\)[/tex] into the Pythagorean identity:

[tex]\[ \sin^2(\theta) + \left(-\frac{2\sqrt{5}}{5}\right)^2 = 1 \][/tex]

2. Square the cosine value:

[tex]\[ \left(-\frac{2\sqrt{5}}{5}\right)^2 = \left(\frac{-2\sqrt{5}}{5}\right) \times \left(\frac{-2\sqrt{5}}{5}\right) = \frac{4 \times 5}{25} = \frac{20}{25} = \frac{4}{5} \][/tex]

3. Substitute [tex]\(\cos^2(\theta)\)[/tex] into the Pythagorean identity:

[tex]\[ \sin^2(\theta) + \frac{4}{5} = 1 \][/tex]

4. Solve for [tex]\(\sin^2(\theta)\)[/tex]:

[tex]\[ \sin^2(\theta) = 1 - \frac{4}{5} \][/tex]

[tex]\[ \sin^2(\theta) = \frac{5}{5} - \frac{4}{5} \][/tex]

[tex]\[ \sin^2(\theta) = \frac{1}{5} \][/tex]

5. Take the square root of both sides to find [tex]\(\sin(\theta)\)[/tex]:

[tex]\[ \sin(\theta) = \pm \sqrt{\frac{1}{5}} = \pm \frac{\sqrt{5}}{5} \][/tex]

6. Determine the correct sign for [tex]\(\sin(\theta)\)[/tex]:

Since [tex]\(\theta\)[/tex] is in quadrant II, where sine is positive, we choose the positive value:

[tex]\[ \sin(\theta) = \frac{\sqrt{5}}{5} \][/tex]

Thus, the correct answer is [tex]\(C\)[/tex]:

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