Sin function for sine waves

You can do sin(input) over a period of time, and use the runtime as an input. This will allow you to receive numbers in a sine pattern. How can I use a pattern such as that to create an auditory sine wave? PWM?

You can do sin(input) over a period of time, and use the runtime as an input. This will allow you to receive numbers in a sine pattern.

This is probably not the best way to go with abn arduino, as it will eat RAM quickly unless you know exactly when to start and stop sampling for your short data set…
What depth & resolution are you planning for the samples?
There may be an altogether better (and probably more complicated) way to do this in a small microcontroller.

How can I use a pattern such as that to create an auditory sine wave? PWM?

PWM - no. Read about PCM for a basic insight to waveform creation.
You’ll need to design or use an external DAC (or function generator) to create decent sine waves.

You can use direct digital synthesis (DDS).

Or you can use an unstable digital filter.

You can use PWM to produce the audio signal or use an arduino Due which has DAC outputs or use an external DAC module.

How can I use a pattern such as that to create an auditory sine wave?

Use the Arduino tone() function with a simple RC low pass filter.