If \( \theta=\frac{-11 \pi}{6} \), then find exact values for the following: \( \sec (\theta) \) equals \( \csc (\theta) \) equals \( \tan (\theta) \) equals \( \cot (\theta) \) equals

Given that \( \theta = \frac{-11\pi}{6} \), we need to find the exact values for the trigonometric functions \( \sec(\theta) \), \( \csc(\theta) \), \( \tan(\theta) \), and \( \cot(\theta) \). First, let's simplify the value of \( \theta \) in terms of radians: \[ \theta = \frac{-11\pi}{6} = -\frac{11\pi}{6} \] Now, we can find the values of the trigonometric functions: 1. \( \sec(\theta) = \frac{1}{\cos(\theta)} \) 2. \( \csc(\theta) = \frac{1}{\sin(\theta)} \) 3. \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \) 4. \( \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)} \) To find the exact values, we need to determine the values of \( \cos(\theta) \), \( \sin(\theta) \), and \( \tan(\theta) \) for \( \theta = -\frac{11\pi}{6} \). Let's calculate these values step by step. Calculate the value by following steps: - step0: Calculate: \(\cos\left(\frac{-11\pi }{6}\right)\) - step1: Rewrite the fraction: \(\cos\left(-\frac{11\pi }{6}\right)\) - step2: Rewrite the expression: \(\cos\left(\frac{11\pi }{6}\right)\) - step3: Calculate the trigonometric value: \(\frac{\sqrt{3}}{2}\) Calculate or simplify the expression \( \sin(-11*\pi/6) \). Calculate the value by following steps: - step0: Calculate: \(\sin\left(\frac{-11\pi }{6}\right)\) - step1: Rewrite the fraction: \(\sin\left(-\frac{11\pi }{6}\right)\) - step2: Rewrite the expression: \(-\sin\left(\frac{11\pi }{6}\right)\) - step3: Evaluate: \(-\left(-\frac{1}{2}\right)\) - step4: Use the rules for multiplication and division: \(\frac{1}{2}\) Calculate or simplify the expression \( \tan(-11*\pi/6) \). Calculate the value by following steps: - step0: Calculate: \(\tan\left(\frac{-11\pi }{6}\right)\) - step1: Rewrite the fraction: \(\tan\left(-\frac{11\pi }{6}\right)\) - step2: Rewrite the expression: \(-\tan\left(\frac{11\pi }{6}\right)\) - step3: Rearrange the numbers: \(-\tan\left(\frac{5\pi }{6}+\pi \right)\) - step4: Rearrange the terms: \(-\tan\left(\frac{5\pi }{6}\right)\) - step5: Evaluate: \(-\left(-\frac{\sqrt{3}}{3}\right)\) - step6: Use the rules for multiplication and division: \(\frac{\sqrt{3}}{3}\) Calculate or simplify the expression \( \cot(-11*\pi/6) \). Calculate the value by following steps: - step0: Calculate: \(\cot\left(\frac{-11\pi }{6}\right)\) - step1: Rewrite the fraction: \(\cot\left(-\frac{11\pi }{6}\right)\) - step2: Rewrite the expression: \(-\cot\left(\frac{11\pi }{6}\right)\) - step3: Rearrange the numbers: \(-\cot\left(\frac{5\pi }{6}+\pi \right)\) - step4: Rearrange the terms: \(-\cot\left(\frac{5\pi }{6}\right)\) - step5: Evaluate: \(-\left(-\sqrt{3}\right)\) - step6: Multiply: \(\sqrt{3}\) Given that \( \theta = -\frac{11\pi}{6} \), we have found the following exact values for the trigonometric functions: 1. \( \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \) 2. \( \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{1}{\frac{1}{2}} = 2 \) 3. \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \) 4. \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} \) Therefore, the exact values for the trigonometric functions are: - \( \sec(\theta) = \frac{2\sqrt{3}}{3} \) - \( \csc(\theta) = 2 \) - \( \tan(\theta) = \frac{\sqrt{3}}{3} \) - \( \cot(\theta) = \sqrt{3} \)