An anharmonic solution to the differential equation describing the oscillations of a simple pendulum at large angles is discussed. The solution is expressed in terms of functions not involving the Jacobi elliptic functions. In the derivation, a sinusoidal expression, including a linear and a Fourier sine series in the argument, has been applied. The coefficients of the Fourier series are found, and it turns out that they are rapidly decreasing. The solution is found to be very close to the exact solution. In the small-angle regime the solution formula for the displacement becomes equivalent to the solution describing the linear pendulum. During the analysis a new formula for the period of the simple pendulum is found, which is more accurate as compared to most previously published results. The formula gives the period with an accuracy better than 0.0004% for angles up to pi/2, and within 0.025% for angles up to 3pi/4 radians. In the small-angle regime the formula becomes equivalent to the result for the period of the linear pendulum. The present derivation of an anharmonic solution to the equation of motion describing a simple pendulum, as well as the derivation of a new expression for the pendulum period, is obtained in terms of elementary functions. It is believed to give valuable insight into a better understanding of the behaviour of the simple pendulum in undergraduate courses on classical mechanics. (Contains 7 figures.)