With examples from Rizzoli & Isles
This is one of those homonym triads that can be really confusing to people, so let's get right down to business and work out the differences between the words. It doesn't help matters that all three can be used as both a verb and a noun. So to look into their differences, we'll just go with the order they were listed in the original question.
Peak has several meanings that tend to do with reaching something very high, a pinnacle of some sort.
- the pointed top of a mountain or ridge; a mountain with a pointed summit; the pointed top of anything. (n)
It was Maura's desire to one day climb to the peak of Mount Kilimanjaro.
- the highest or most important point or level. (n)
For Jane, being a detective would always be the peak of her career.
- the maximum point, degree, or volume of anything. (n)
"Crime rates hit their peak around the holidays, Frost," Korsak said. "It's just a fact of life."
- to attain a peak of activity, development, popularity, etc. (v)
"No, Jane, you don't understand," Maura explained. "These shoes haven't even peaked in popularity yet, but they will."
- being at the point of maximum frequency, intensity, use, etc.; busiest or most active; constituting the highest or maximum level, volume, etc.; optimal; prime. (adj)
"So you're saying that this Boston Strangler copycat is running rampant during the peak tourist season?" Frankie asked his sister.
Pique, used for excitation of the emotions, is mostly seen in its verb form, but it can also be used as a noun.
- to affect with sharp irritation and resentment, especially by some wound to pride; to wound (the pride, vanity, etc.). (v)
"It piqued you grievously?" Jane asked, staring at Maura. "Why can't you just say that it pissed you off or upset you?"
- to excite (interest, curiosity, etc.). (v)
Jane would never publicly admit that Maura's enthusiasm for expensive shoes piqued her interest.
- a feeling of irritation or resentment, as from a wound to pride or self-esteem. (n)
"Admit it, Maura. You were jealous."
"I will do no such thing," Maura replied. "I was not in a pique because Bass preferred your treat to mine."
Peek usually involves looking at something.
- to look or glance quickly or furtively, especially through a small opening or from a concealed location; peep; peer. (v)
Jane peeked through the keyhole after she heard Maura making a strange sound in the bathroom.
- a quick or furtive look or glance; peep. (n)
"Hey, Janey, can you take a peek at this for me?" Korsak asked, sounding disgusted. "I left my damned glasses on my desk."
So a good way to remember how these words are used would be as follows: if you have to reach for it, use peak; if you can see it, use peek; if it involves the emotions, use pique.