Define the rms value of an alternating current. Show that it is 0.707 times the peak value.

How to lay out the answer. Follow the frame, then write it in your own words.

1 · Define. The rms value is the steady current that produces the same heat in a resistor in the same time as the AC.
2 · Write the power. i = iₘ sin ωt, so p = i²R = iₘ²R sin² ωt.
3 · Take the average. The average of sin² ωt over a cycle is ½, so the average power is ½iₘ²R.
4 · Equate. I²R = ½iₘ²R gives I = iₘ/√2 = 0.707 iₘ.
5 · Real life. 220 V mains is an rms value; its peak is about 311 V.
6 · Close in one line. So, the rms current is 70.7% of the peak current, and likewise V = vₘ/√2.

Scoring tip: Show why the average of sin² ωt is ½, using sin² ωt = ½(1 − cos 2ωt).