Sleep cycles are 90 minutes long. Sleeping in 90 minute increments will help you sleep efficiently and allow you to wake up feeling refreshed.  There’s even an app for that! 
Sleep cycles are comprised of 5 stages; stages 1-4 and REM. While it is true the mean sleep cycle is 90 minutes, the range is actually 60 – 110 minutes. Journalistic enthusiasm seems to have glossed over the simple, but critical, fact that 90 minutes is only an average length.  Additionally, after the first few sleep cycles, that amount of time you spend in stages 3 and 4 decreases, while the time spent in stages 1, 2 and REM increases. 
Since it’s almost impossible to time your sleep cycles, sleeping in 90 minute increments does not ensure you will wake up refreshed. That is why waking up to a traditional alarm is like playing Russian Roulette with your early morning sanity. 
However, waking up at the end of a sleep cycle does help you sleep efficiently and wake up refreshed. It is just hard to do so without outside help…
How the WakeMate can help:
We built the WakeMate to adapt to the unpredictability of your sleep. The best way to achieve a better night’s rest is to monitor your sleeping patterns over the course of the night. The WakeMate monitors a person’s sleep cycles via Actigraphy, a well-respected scientific monitoring system that has been employed by sleep labs for decades. 
The graph above is a sneak preview of a feature we are currently working on aimed at helping you figure out how long to sleep. It gives you real data on your personal sleep cycles. The x-axis represents how long you have been asleep and the y-axis represents the likelihood you are in each stage of sleep for any given minute. We calculate this by analyzing your WakeMate sleep history. For example, 4 hours after I fall asleep there is a ~40% chance I will be in deep sleep and a ~20% chance I will be in light sleep.
From the data above, you can see that at 6:45 I switch from a deep sleep cycle to a light sleep cycle. The same thing also happens at 8:00. Therefore, in order to maximize the amount of deep sleep I experience and wake up at the end of a sleep cycle, I should sleep for 6 hours, 45 minutes or 8 hours.
To feel refreshed when your alarm goes off, the best times to wake-up are at the end of a sleep cycle or during “almost awake” points which occur during REM sleep. These “almost awake” points are certain moments during REM when your brain thinks you are awake even though you are actually dreaming. The WakeMate finds these points using Actigraphy by analyzing the frequency and amplitude of your movements. The WakeMate then sounds the alarm so the transition to wakefulness is smooth and the urge to smash your alarm clock is not as strong.
tl;dr: Even if sleeping in 90 minute increments is not a useful strategy, the WakeMate is here to ensure you get the most out of your sleep and wake up feeling refreshed.
 Tassi P., & Muzet, A. (2000). Sleep inertia. Sleep Medicine Reviews, 4, 341-353.