Answer :
To determine where the function \(\tan \theta\) is undefined, we need to recall the property of the tangent function. The tangent function, \(\tan \theta\), can be expressed as:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \][/tex]
This ratio becomes undefined wherever \(\cos \theta = 0\), because dividing by zero is undefined.
To identify these points, let's think about the values of \(\theta\) where \(\cos \theta = 0\). The cosine function \(\cos \theta\) equals zero at odd multiples of \(\frac{\pi}{2}\), i.e.,
[tex]\[ \theta = \frac{(2n+1)\pi}{2} \quad \text{for any integer } n. \][/tex]
Let's evaluate each option:
A. \(\theta = \pi\)
[tex]\[ \cos \pi = -1 \quad \Rightarrow \tan \pi = \frac{\sin \pi}{\cos \pi} = \frac{0}{-1} = 0 \][/tex]
Since \(\cos \pi \neq 0\), \(\tan \pi\) is not undefined.
B. \(\theta = \frac{3\pi}{2}\)
[tex]\[ \cos \frac{3\pi}{2} = 0 \quad \Rightarrow \tan \frac{3\pi}{2} = \frac{\sin \frac{3\pi}{2}}{\cos \frac{3\pi}{2}} = \frac{-1}{0} \][/tex]
Since \cos\left(\frac{3\pi}{2}\right) = 0, \(\tan \frac{3\pi}{2}\) is undefined.
C. \(\theta = \frac{\pi}{2}\)
[tex]\[ \cos \frac{\pi}{2} = 0 \quad \Rightarrow \tan \frac{\pi}{2} = \frac{\sin \frac{\pi}{2}}{\cos \frac{\pi}{2}} = \frac{1}{0} \][/tex]
Since \cos\left(\frac{\pi}{2}\right) = 0, \(\tan \frac{\pi}{2}\) is undefined.
D. \(\theta = 0\)
[tex]\[ \cos 0 = 1 \quad \Rightarrow \tan 0 = \frac{\sin 0}{\cos 0} = \frac{0}{1} = 0 \][/tex]
Since \cos 0 \neq 0, \(\tan 0\) is not undefined.
Thus, the values of \(\theta\) for which \(\tan \theta\) is undefined are:
[tex]\[ B. \frac{3 \pi}{2} \][/tex]
[tex]\[ C. \frac{\pi}{2} \][/tex]
These correspond to options B and C.
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \][/tex]
This ratio becomes undefined wherever \(\cos \theta = 0\), because dividing by zero is undefined.
To identify these points, let's think about the values of \(\theta\) where \(\cos \theta = 0\). The cosine function \(\cos \theta\) equals zero at odd multiples of \(\frac{\pi}{2}\), i.e.,
[tex]\[ \theta = \frac{(2n+1)\pi}{2} \quad \text{for any integer } n. \][/tex]
Let's evaluate each option:
A. \(\theta = \pi\)
[tex]\[ \cos \pi = -1 \quad \Rightarrow \tan \pi = \frac{\sin \pi}{\cos \pi} = \frac{0}{-1} = 0 \][/tex]
Since \(\cos \pi \neq 0\), \(\tan \pi\) is not undefined.
B. \(\theta = \frac{3\pi}{2}\)
[tex]\[ \cos \frac{3\pi}{2} = 0 \quad \Rightarrow \tan \frac{3\pi}{2} = \frac{\sin \frac{3\pi}{2}}{\cos \frac{3\pi}{2}} = \frac{-1}{0} \][/tex]
Since \cos\left(\frac{3\pi}{2}\right) = 0, \(\tan \frac{3\pi}{2}\) is undefined.
C. \(\theta = \frac{\pi}{2}\)
[tex]\[ \cos \frac{\pi}{2} = 0 \quad \Rightarrow \tan \frac{\pi}{2} = \frac{\sin \frac{\pi}{2}}{\cos \frac{\pi}{2}} = \frac{1}{0} \][/tex]
Since \cos\left(\frac{\pi}{2}\right) = 0, \(\tan \frac{\pi}{2}\) is undefined.
D. \(\theta = 0\)
[tex]\[ \cos 0 = 1 \quad \Rightarrow \tan 0 = \frac{\sin 0}{\cos 0} = \frac{0}{1} = 0 \][/tex]
Since \cos 0 \neq 0, \(\tan 0\) is not undefined.
Thus, the values of \(\theta\) for which \(\tan \theta\) is undefined are:
[tex]\[ B. \frac{3 \pi}{2} \][/tex]
[tex]\[ C. \frac{\pi}{2} \][/tex]
These correspond to options B and C.
