Of course! Let's determine the values of [tex]\(\sin x\)[/tex] and [tex]\(\cos x\)[/tex] given that [tex]\(\sin x = \frac{3}{5}\)[/tex] and [tex]\(x\)[/tex] lies in the second quadrant. We'll use trigonometric identities to find these values.
### Step-by-Step Solution:
1. Identify [tex]\(\sin x\)[/tex]:
[tex]\[
\sin x = \frac{3}{5}
\][/tex]
We are given that [tex]\(\sin x\)[/tex] is [tex]\(\frac{3}{5}\)[/tex], which indicates the ratio of the opposite side to the hypotenuse in a right-angled triangle.
2. Determine the quadrant:
Since [tex]\(x\)[/tex] lies in the second quadrant, we know that:
- [tex]\(\sin x\)[/tex] is positive.
- [tex]\(\cos x\)[/tex] is negative.
3. Use the Pythagorean Identity:
The Pythagorean identity for sine and cosine is:
[tex]\[
\sin^2 x + \cos^2 x = 1
\][/tex]
Substitute [tex]\(\sin x = \frac{3}{5}\)[/tex] into the identity:
[tex]\[
\left(\frac{3}{5}\right)^2 + \cos^2 x = 1
\][/tex]
4. Solve for [tex]\(\cos^2 x\)[/tex]:
Simplify the squared sine term:
[tex]\[
\left(\frac{3}{5}\right)^2 = \frac{9}{25}
\][/tex]
Substitute this value into the identity:
[tex]\[
\frac{9}{25} + \cos^2 x = 1
\][/tex]
Rearrange the equation to solve for [tex]\(\cos^2 x\)[/tex]:
[tex]\[
\cos^2 x = 1 - \frac{9}{25}
\][/tex]
[tex]\[
\cos^2 x = \frac{25}{25} - \frac{9}{25}
\][/tex]
[tex]\[
\cos^2 x = \frac{16}{25}
\][/tex]
5. Determine [tex]\(\cos x\)[/tex]:
Take the square root of both sides to find [tex]\(\cos x\)[/tex]:
[tex]\[
\cos x = \pm \sqrt{\frac{16}{25}}
\][/tex]
[tex]\[
\cos x = \pm \frac{4}{5}
\][/tex]
Since we are in the second quadrant where cosine is negative:
[tex]\[
\cos x = -\frac{4}{5}
\][/tex]
### Conclusion:
Given that [tex]\(\sin x = \frac{3}{5}\)[/tex] and [tex]\(x\)[/tex] lies in the second quadrant, we find:
[tex]\[
\sin x = \frac{3}{5} \quad \text{(already given)}
\][/tex]
[tex]\[
\cos x = -\frac{4}{5}
\][/tex]
Thus, the values are:
[tex]\[
\sin x = \frac{3}{5}, \quad \cos x = -\frac{4}{5}
\][/tex]
