The main reason to use the Arduino is to use a D-to-A converter to generate an output wave by generating the sampled output. A sampled sin wave is actually output by the Arduino Script program listed in Source 07. Many more images and some additional programming information are included in Source 01.
The 5 minute video in Source 02 explains how to drive an MCP4725 DAC (US$ 1.00) using an Arduino. It does not show any oscilloscope screens nor any Arduino code.
The nearly 1 hour videos (probably by Peter of "thebreadboard" videos or Peter Oakes of www.thebreadboard.ca ) in Sources 03 and 04 explain how to drive an MCP4725 DAC and an ADS1220 (US$ 1.00) using an Arduino. These two devices are both quad devices. He also explains how to solder a tiny IC (in a TSSOP-16 package) using a hot air gun and alcohol wipes. He then solders on the necessary headers.
ADC
The author of the video uses an ADS1220 for A-to-D converstion. See Source 08 which is the data sheet for the ti ADS1220.
DAC using I2C
He uses a DAC8574 (into a MSP430F449) for D-to-A. The DAC I2C address is 098h which must be shifted left 1 bit (to become 04Ch). Channel 4 outputs the correct signal. He refers to a TI PDF Application Note that is available in Source 05 for the DAC.
Later in the 2nd video he uses an Arduino script to create a sin wave. His Arduino Script code (corrected) in txt mode is in Source 07. A zipped version is in Source 06. This code produces a "sampled" version of an analog sine wave on the DAC8574 channel 4 output pin.