To find [tex]\(\cos(\theta)\)[/tex] using the Pythagorean identity given that [tex]\(\sin(\theta) = -\frac{3}{4}\)[/tex] and the angle [tex]\(\theta\)[/tex] is in the second quadrant, follow these steps:
1. Apply the Pythagorean Identity:
The Pythagorean identity states:
[tex]\[
\sin^2(\theta) + \cos^2(\theta) = 1
\][/tex]
2. Square the given value of [tex]\(\sin(\theta)\)[/tex]:
Given [tex]\(\sin(\theta) = -\frac{3}{4}\)[/tex], we calculate:
[tex]\[
\sin^2(\theta) = \left(-\frac{3}{4}\right)^2 = \frac{9}{16}
\][/tex]
3. Substitute [tex]\(\sin^2(\theta)\)[/tex] into the Pythagorean identity:
Substitute [tex]\(\sin^2(\theta) = \frac{9}{16}\)[/tex] into the identity:
[tex]\[
\frac{9}{16} + \cos^2(\theta) = 1
\][/tex]
4. Solve for [tex]\(\cos^2(\theta)\)[/tex]:
Rearrange the equation to solve for [tex]\(\cos^2(\theta)\)[/tex]:
[tex]\[
\cos^2(\theta) = 1 - \frac{9}{16}
\][/tex]
Since [tex]\(1\)[/tex] can be written as [tex]\(\frac{16}{16}\)[/tex], we get:
[tex]\[
\cos^2(\theta) = \frac{16}{16} - \frac{9}{16} = \frac{7}{16}
\][/tex]
5. Take the square root to find [tex]\(\cos(\theta)\)[/tex]:
To find [tex]\(\cos(\theta)\)[/tex], take the square root of both sides:
[tex]\[
\cos(\theta) = \pm \sqrt{\frac{7}{16}} = \pm \frac{\sqrt{7}}{4}
\][/tex]
6. Determine the correct sign of [tex]\(\cos(\theta)\)[/tex]:
Since [tex]\(\theta\)[/tex] is in the second quadrant, where cosine is negative, we select the negative value:
[tex]\[
\cos(\theta) = -\frac{\sqrt{7}}{4}
\][/tex]
Hence, the exact and simplified value of [tex]\(\cos(\theta)\)[/tex] is:
[tex]\[
\cos(\theta) = -\frac{ \sqrt{7}}{4}
\][/tex]
