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Answer :
Sure! Let's find [tex]\(\cos(\theta)\)[/tex] given that [tex]\(\sin(\theta) = \frac{24}{25}\)[/tex] and [tex]\(\theta\)[/tex] is in the second quadrant.
1. Understand the Quadrant: In the second quadrant, sine is positive and cosine is negative.
2. Use the Pythagorean Identity: The Pythagorean identity tells us that:
[tex]\[
\sin^2(\theta) + \cos^2(\theta) = 1
\][/tex]
Given [tex]\(\sin(\theta) = \frac{24}{25}\)[/tex], we can find [tex]\(\sin^2(\theta)\)[/tex]:
[tex]\[
\sin^2(\theta) = \left(\frac{24}{25}\right)^2 = \frac{576}{625}
\][/tex]
3. Solve for [tex]\(\cos^2(\theta)\)[/tex]:
[tex]\[
\cos^2(\theta) = 1 - \sin^2(\theta) = 1 - \frac{576}{625} = \frac{49}{625}
\][/tex]
4. Find [tex]\(\cos(\theta)\)[/tex]: Take the square root of [tex]\(\cos^2(\theta)\)[/tex]:
[tex]\[
\cos(\theta) = \pm \sqrt{\frac{49}{625}} = \pm \frac{7}{25}
\][/tex]
5. Determine the Sign of [tex]\(\cos(\theta)\)[/tex]: Since [tex]\(\theta\)[/tex] is in the second quadrant and cosine is negative there:
[tex]\[
\cos(\theta) = -\frac{7}{25}
\][/tex]
So, the value of [tex]\(\cos(\theta)\)[/tex] is [tex]\(-\frac{7}{25}\)[/tex].
1. Understand the Quadrant: In the second quadrant, sine is positive and cosine is negative.
2. Use the Pythagorean Identity: The Pythagorean identity tells us that:
[tex]\[
\sin^2(\theta) + \cos^2(\theta) = 1
\][/tex]
Given [tex]\(\sin(\theta) = \frac{24}{25}\)[/tex], we can find [tex]\(\sin^2(\theta)\)[/tex]:
[tex]\[
\sin^2(\theta) = \left(\frac{24}{25}\right)^2 = \frac{576}{625}
\][/tex]
3. Solve for [tex]\(\cos^2(\theta)\)[/tex]:
[tex]\[
\cos^2(\theta) = 1 - \sin^2(\theta) = 1 - \frac{576}{625} = \frac{49}{625}
\][/tex]
4. Find [tex]\(\cos(\theta)\)[/tex]: Take the square root of [tex]\(\cos^2(\theta)\)[/tex]:
[tex]\[
\cos(\theta) = \pm \sqrt{\frac{49}{625}} = \pm \frac{7}{25}
\][/tex]
5. Determine the Sign of [tex]\(\cos(\theta)\)[/tex]: Since [tex]\(\theta\)[/tex] is in the second quadrant and cosine is negative there:
[tex]\[
\cos(\theta) = -\frac{7}{25}
\][/tex]
So, the value of [tex]\(\cos(\theta)\)[/tex] is [tex]\(-\frac{7}{25}\)[/tex].
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