Find the exact value of each of the remaining trigonometric functions of $\theta$. \[ \sin \theta=1 \] Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $\cos \theta=$ (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) B. The function is undefined. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $\tan \theta=$ (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) B. The function is undefined. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $\csc \theta=$ (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) B. The function is undefined. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $\sec \theta=$ (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) B. The function is undefined.
Solution
Step 1 ：Given that \(\sin \theta = 1\), we know that \(\theta\) is \(90^\circ\) or \(\frac{\pi}{2}\) radians.
Step 2 ：Using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\), we can solve for \(\cos \theta\). Substituting \(\sin \theta = 1\) into the equation, we get \(1 + \cos^2 \theta = 1\). Solving for \(\cos \theta\), we get \(\cos \theta = 0\). So, \(\cos \theta = \boxed{0}\).
Step 3 ：Next, we find \(\tan \theta\) using the identity \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Substituting \(\sin \theta = 1\) and \(\cos \theta = 0\) into the equation, we see that \(\tan \theta\) is undefined because we cannot divide by zero. So, \(\tan \theta = \text{undefined}\).
Step 4 ：Next, we find \(\csc \theta\) using the identity \(\csc \theta = \frac{1}{\sin \theta}\). Substituting \(\sin \theta = 1\) into the equation, we get \(\csc \theta = 1\). So, \(\csc \theta = \boxed{1}\).
Step 5 ：Finally, we find \(\sec \theta\) using the identity \(\sec \theta = \frac{1}{\cos \theta}\). Substituting \(\cos \theta = 0\) into the equation, we see that \(\sec \theta\) is undefined because we cannot divide by zero. So, \(\sec \theta = \text{undefined}\).
