Let Theta be an angle such that secTheta =(5)/(4) and tanTheta lt 0 Find the exact values of cotTheta and cscTheta cotTheta =square cscTheta =square

To find the exact values of $\cot\Theta$ and $\csc\Theta$, we can use the given information that $\sec\Theta = \frac{5}{4}$ and $\tan\Theta < 0$.<br /><br />First, let's find $\cos\Theta$ using the reciprocal identity of secant:<br />$\sec\Theta = \frac{1}{\cos\Theta}$<br />$\frac{5}{4} = \frac{1}{\cos\Theta}$<br />$\cos\Theta = \frac{4}{5}$<br /><br />Since $\tan\Theta < 0$, we know that $\Theta$ is in the second or fourth quadrant. In the second quadrant, cosine is positive and sine is positive, while in the fourth quadrant, cosine is positive and sine is negative.<br /><br />Now, let's find $\sin\Theta$ using the Pythagorean identity:<br />$\sin^2\Theta + \cos^2\Theta = 1$<br />$\sin^2\Theta + \left(\frac{4}{5}\right)^2 = 1$<br />$\sin^2\Theta + \frac{16}{25} = 1$<br />$\sin^2\Theta = \frac{9}{25}$<br />$\sin\Theta = \pm\frac{3}{5}$<br /><br />Since $\Theta$ is in the second or fourth quadrant, we need to determine the sign of $\sin\Theta$. In the second quadrant, sine is positive, so $\sin\Theta = \frac{3}{5}$. In the fourth quadrant, sine is negative, so $\sin\Theta = -\frac{3}{5}$.<br /><br />Now, let's find $\cot\Theta$ using the reciprocal identity of tangent:<br />$\cot\Theta = \frac{1}{\tan\Theta}$<br />Since $\tan\Theta = \frac{\sin\Theta}{\cos\Theta}$, we have:<br />$\cot\Theta = \frac{\cos\Theta}{\sin\Theta}$<br />$\cot\Theta = \frac{\frac{4}{5}}{\pm\frac{3}{5}}$<br />$\cot\Theta = \pm\frac{4}{3}$<br /><br />Since $\Theta$ is in the second or fourth quadrant, we need to determine the sign of $\cot\Theta$. In the second quadrant, cotangent is negative, so $\cot\Theta = -\frac{4}{3}$. In the fourth quadrant, cotangent is positive, so $\cot\Theta = \frac{4}{3}$.<br /><br />Finally, let's find $\csc\Theta$ using the reciprocal identity of sine:<br />$\csc\Theta = \frac{1}{\sin\Theta}$<br />$\csc\Theta = \frac{1}{\pm\frac{3}{5}}$<br />$\csc\Theta = \pm\frac{5}{3}$<br /><br />Since $\Theta$ is in the second or fourth quadrant, we need to determine the sign of $\csc\Theta$. In the second quadrant, cosecant is positive, so $\csc\Theta = \frac{5}{3}$. In the fourth quadrant, cosecant is negative, so $\csc\Theta = -\frac{5}{3}$.<br /><br />Therefore, the exact values of $\cot\Theta$ and $\csc\Theta$ are:<br />$\cot\Theta = -\frac{4}{3}$<br />$\csc\Theta = \frac{5}{3}$