Get the best answers to your questions with the help of IDNLearn.com's experts. Discover in-depth answers from knowledgeable professionals, providing you with the information you need.
Sagot :
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]
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. Thank you for visiting IDNLearn.com. We’re here to provide clear and concise answers, so visit us again soon.