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]
[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]
