Question 11 of 29 Find the exact value of each of the remaining trigonometric functions of $\theta$. \[ \sin \theta=\frac{1}{2}, \cot \theta<0 \] \[ \cos \theta= \] (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) \[ \cot \theta= \] (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) \[ \tan \theta= \] (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) \[ \csc \theta= \] (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) \[ \sec \theta= \] (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
Solution
Step 1 ：Given that \(\sin \theta = \frac{1}{2}\) and \(\cot \theta < 0\), we can determine the quadrant in which \(\theta\) lies. Since \(\sin \theta > 0\) and \(\cot \theta < 0\), \(\theta\) must be in the second quadrant.
Step 2 ：Next, we can find the value of \(\cos \theta\) using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\). Solving for \(\cos \theta\), we get \(\cos \theta = \sqrt{1 - \sin^2 \theta} = \sqrt{1 - \left(\frac{1}{2}\right)^2} = \sqrt{1 - \frac{1}{4}} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}\). However, since \(\theta\) is in the second quadrant, \(\cos \theta\) is negative. Therefore, \(\cos \theta = -\frac{\sqrt{3}}{2}\).
Step 3 ：Then, we can find the value of \(\tan \theta\) using the identity \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Substituting the given values, we get \(\tan \theta = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = -\frac{1}{\sqrt{3}}\).
Step 4 ：Next, we can find the value of \(\cot \theta\) using the identity \(\cot \theta = \frac{1}{\tan \theta}\). Substituting the value of \(\tan \theta\) we found, we get \(\cot \theta = \frac{1}{-\frac{1}{\sqrt{3}}} = -\sqrt{3}\).
Step 5 ：Then, we can find the value of \(\csc \theta\) using the identity \(\csc \theta = \frac{1}{\sin \theta}\). Substituting the given value of \(\sin \theta\), we get \(\csc \theta = \frac{1}{\frac{1}{2}} = 2\).
Step 6 ：Finally, we can find the value of \(\sec \theta\) using the identity \(\sec \theta = \frac{1}{\cos \theta}\). Substituting the value of \(\cos \theta\) we found, we get \(\sec \theta = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}\).
Step 7 ：So, the exact values of the remaining trigonometric functions of \(\theta\) are \(\cos \theta = -\frac{\sqrt{3}}{2}\), \(\tan \theta = -\frac{1}{\sqrt{3}}\), \(\cot \theta = -\sqrt{3}\), \(\csc \theta = 2\), and \(\sec \theta = -\frac{2\sqrt{3}}{3}\).
