Suppose that f is a function of the type of cos x = cos 17 x which of the following functions as he has a property sin x is equal to sin 17 x

To explore the functions with the property $\sin x = \sin 17x$, we need to analyze the implications of this equality in terms of periodicity and symmetry of the sine function.

1. Understanding the Property of Sine Function: The sine function, $\sin x$, has a periodicity of $2\pi$. Hence, the equation $\sin x = \sin 17x$ will be satisfied when: $x = 17x + 2k\pi \quad \text{or} \quad x = \pi - 17x + 2k\pi$ for some integer $k$.

2. Solving the Equation:

   • From $x = 17x + 2k\pi$: $x - 17x = 2k\pi \implies -16x = 2k\pi \implies x = -\frac{k\pi}{8}$
   • From $x = \pi - 17x + 2k\pi$: $x + 17x = \pi + 2k\pi \implies 18x = (2k + 1)\pi \implies x = \frac{(2k + 1)\pi}{18}$

3. Verification and Periodicity: We can summarize our findings:

   • The solutions $x = -\frac{k\pi}{8}$ imply an infinite number of solutions as $k$ varies.
   • The solutions $x = \frac{(2k + 1)\pi}{18}$ also give an infinite number of solutions as $k$ varies.

4. Final Functions: Thus, the functions that satisfy the property $\sin x = \sin 17 x$ can be expressed as: $x = -\frac{k\pi}{8} \quad \text{and} \quad x = \frac{(2k + 1)\pi}{18}$ where $k$ is an integer.

Final Answer

The functions that satisfy $\sin x = \sin 17 x$ are $x = -\frac{k\pi}{8}$ and $x = \frac{(2k + 1)\pi}{18}$ for integer values of $k$.