Given that sinTheta =(3)/(7) where Theta is in the second quadrant determine the exact value of secTheta and cotTheta

## Step 1:<br />Given that \( \sin\Theta = \frac{3}{7} \) and \(\Theta\) is in the second quadrant.<br /><br />## Step 2:<br />Recall that in the second quadrant, sine is positive, and cosine is negative. We need to find \(\cos\Theta\).<br /><br />## Step 3:<br />Use the Pythagorean identity: <br />### \( \sin^2\Theta + \cos^2\Theta = 1 \)<br /><br />Substitute \( \sin\Theta = \frac{3}{7} \):<br />### \( \left( \frac{3}{7} \right)^2 + \cos^2\Theta = 1 \)<br /><br />## Step 4:<br />Calculate \( \left( \frac{3}{7} \right)^2 \):<br />### \( \frac{9}{49} \)<br /><br />## Step 5:<br />Solve for \( \cos^2\Theta \):<br />### \( \cos^2\Theta = 1 - \frac{9}{49} \)<br />### \( \cos^2\Theta = \frac{49}{49} - \frac{9}{49} \)<br />### \( \cos^2\Theta = \frac{40}{49} \)<br /><br />## Step 6:<br />Take the square root of both sides, considering the sign in the second quadrant:<br />### \( \cos\Theta = -\sqrt{\frac{40}{49}} \)<br />### \( \cos\Theta = -\frac{\sqrt{40}}{7} \)<br />### \( \cos\Theta = -\frac{2\sqrt{10}}{7} \)<br /><br />## Step 7:<br />Find \( \sec\Theta \), which is the reciprocal of \( \cos\Theta \):<br />### \( \sec\Theta = \frac{1}{\cos\Theta} \)<br />### \( \sec\Theta = \frac{1}{-\frac{2\sqrt{10}}{7}} \)<br />### \( \sec\Theta = -\frac{7}{2\sqrt{10}} \)<br /><br />## Step 8:<br />Find \( \cot\Theta \), which is \( \frac{\cos\Theta}{\sin\Theta} \):<br />### \( \cot\Theta = \frac{-\frac{2\sqrt{10}}{7}}{\frac{3}{7}} \)<br />### \( \cot\Theta = -\frac{2\sqrt{10}}{3} \)<br /><br />Thus, the exact values are:<br />## \( \sec\Theta = -\frac{7}{2\sqrt{10}} \)<br />## \( \cot\Theta = -\frac{2\sqrt{10}}{3} \)