To determine which functions have no zeros, let's analyze each function:
-
\(y = \cos x\): This function has zeros at \(x = \frac{\pi}{2} + n\pi\) (where \(n\) is any integer). So, it has zeros.
-
\(y = \cot x\): The cotangent function is \( \frac{\cos x}{\sin x} \) and has zeros at \(x = n\pi\), where \(n\) is any integer. So, it has zeros.
-
\(y = \csc x\): The cosecant function is the reciprocal of sine, \( \frac{1}{\sin x} \), and it has undefined values (asymptotes) when \(\sin x = 0\) (i.e., at \(x = n\pi\)), but it does not have zeros. It is always defined (except at multiples of \(\pi\)) and never equals zero.
-
\(y = \sec x\): The secant function is the reciprocal of cosine, \( \frac{1}{\cos x} \), and is undefined when \(\cos x = 0\) (i.e., at \(x = \frac{\pi}{2} + n\pi\)), but it does not have zeros. It is always defined (except at odd multiples of \(\pi/2\)) and never equals zero.
-
\(y = \sin x\): This function has zeros at \(x = n\pi\), where \(n\) is any integer. So, it has zeros.
-
\(y = \tan x\): The tangent function is \(\frac{\sin x}{\cos x}\) and has zeros at \(x = n\pi\), where \(n\) is any integer. So, it has zeros.
Summary: The functions that have no zeros are:
- \(y = \csc x\)
- \(y = \sec x\)
Thus, the correct responses are:
- \(y = \csc x\)
- \(y = \sec x\)
1 answer