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Answer :
To find [tex]\(\cos(\theta)\)[/tex] given that [tex]\(\sin(\theta) = \frac{24}{25}\)[/tex] and [tex]\(\theta\)[/tex] is in Quadrant II, we can use the Pythagorean identity and properties of trigonometric functions in different quadrants. Here's how you can do it step-by-step:
1. Understand the Pythagorean Identity:
The Pythagorean identity states that for any angle [tex]\(\theta\)[/tex]:
[tex]\[
\sin^2(\theta) + \cos^2(\theta) = 1
\][/tex]
Since we know [tex]\(\sin(\theta) = \frac{24}{25}\)[/tex], we can use this identity to find [tex]\(\cos(\theta)\)[/tex].
2. Substitute the Given Value and Solve for [tex]\(\cos^2(\theta)\)[/tex]:
Substitute [tex]\(\sin(\theta) = \frac{24}{25}\)[/tex] into the identity:
[tex]\[
\left(\frac{24}{25}\right)^2 + \cos^2(\theta) = 1
\][/tex]
Calculate [tex]\(\left(\frac{24}{25}\right)^2\)[/tex]:
[tex]\[
\frac{24}{25} \times \frac{24}{25} = \frac{576}{625}
\][/tex]
Substitute back into the equation:
[tex]\[
\frac{576}{625} + \cos^2(\theta) = 1
\][/tex]
3. Isolate [tex]\(\cos^2(\theta)\)[/tex]:
[tex]\[
\cos^2(\theta) = 1 - \frac{576}{625}
\][/tex]
Convert 1 to a fraction with a denominator of 625:
[tex]\[
1 = \frac{625}{625}
\][/tex]
Subtract:
[tex]\[
\cos^2(\theta) = \frac{625}{625} - \frac{576}{625} = \frac{49}{625}
\][/tex]
4. Take the Square Root:
To 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]:
In Quadrant II, the cosine of any angle is negative (since [tex]\(x\)[/tex]-coordinates are negative in that quadrant). Therefore, [tex]\(\cos(\theta)\)[/tex] is:
[tex]\[
\cos(\theta) = -\frac{7}{25}
\][/tex]
Thus, [tex]\(\cos(\theta) = -\frac{7}{25}\)[/tex].
1. Understand the Pythagorean Identity:
The Pythagorean identity states that for any angle [tex]\(\theta\)[/tex]:
[tex]\[
\sin^2(\theta) + \cos^2(\theta) = 1
\][/tex]
Since we know [tex]\(\sin(\theta) = \frac{24}{25}\)[/tex], we can use this identity to find [tex]\(\cos(\theta)\)[/tex].
2. Substitute the Given Value and Solve for [tex]\(\cos^2(\theta)\)[/tex]:
Substitute [tex]\(\sin(\theta) = \frac{24}{25}\)[/tex] into the identity:
[tex]\[
\left(\frac{24}{25}\right)^2 + \cos^2(\theta) = 1
\][/tex]
Calculate [tex]\(\left(\frac{24}{25}\right)^2\)[/tex]:
[tex]\[
\frac{24}{25} \times \frac{24}{25} = \frac{576}{625}
\][/tex]
Substitute back into the equation:
[tex]\[
\frac{576}{625} + \cos^2(\theta) = 1
\][/tex]
3. Isolate [tex]\(\cos^2(\theta)\)[/tex]:
[tex]\[
\cos^2(\theta) = 1 - \frac{576}{625}
\][/tex]
Convert 1 to a fraction with a denominator of 625:
[tex]\[
1 = \frac{625}{625}
\][/tex]
Subtract:
[tex]\[
\cos^2(\theta) = \frac{625}{625} - \frac{576}{625} = \frac{49}{625}
\][/tex]
4. Take the Square Root:
To 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]:
In Quadrant II, the cosine of any angle is negative (since [tex]\(x\)[/tex]-coordinates are negative in that quadrant). Therefore, [tex]\(\cos(\theta)\)[/tex] is:
[tex]\[
\cos(\theta) = -\frac{7}{25}
\][/tex]
Thus, [tex]\(\cos(\theta) = -\frac{7}{25}\)[/tex].
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