Common mitre saws, mitre boxes or mitre gauges will not cut a 60-degree angle -- it's too sharp. Some mitre saws will cut up to 50 degrees, but they are limited by the way they are designed. You can cut a 60-degree angle by hand, but that is inaccurate and sloppy. It's easy to cut a 60-degree angle on a common mitre saw if you use a trick that craftsmen have been using for years.
Draw the 60-degree angle on the board using a protractor and a pencil. Try to imagine where the saw blade will cut through the wood and draw the line where you want the saw blade to cut.
Set the board on the mitre saw against the fence with the line on the left side of the blade. Swing the blade over to the line. It won't go far enough -- just push it as far as it will go and line up the blade with the line as close as you can get it.
Tap a wedge between the board and the mitre saw fence with a hammer on the far left outer corner of the mitre saw fence. This will tilt the angle of the board sharper. Pull the handle of the saw down -- don't turn on the saw -- and lean down and sight down the blade and the line at the same time.
Tape on the wedge as you watch the angle get sharper as you tap. When the blade and the line align, stop tapping the wedge.
Hold the wood against the fence with your left hand. Hold it securely to pin the wedge against the fence maintaining the angle. Place your right hand on the saw handle, pull the trigger to turn on the saw and bring the blade down to cut the 60-degree angle.
You can ease the blade down to make a shallow test cut. Use the protractor to measure the angle. If sufficient, make the final cut the same way.
Always wear safety glasses and breathing protection when working with wood.