The usage of sigh
Solution 1:
the dictionary in its examples reads:
sigh
Verb (used without object)
to let out one's breath audibly, as from sorrow, weariness, or relief.
Now relief is a synonym of contentment
relief
Synonyms for relief
alleviation assistance comfort happiness help maintenance reprieve respite satisfaction support abatement amelioration appeasement assuagement balm break breather cheer consolation contentment cure deliverance diversion ease easement extrication fix hand letup lift lightening mitigation mollification palliative refreshment release remission rest restfulness softening solace succor sustenance allayment comforting load off one's mind quick fix
dictionary.com
Furthermore drawing a deep breath holds a different meaning.
- draw breath
: to pause for a moment to take a breath or breathe more slowly
cambridge.org
Your teacher is clearly wrong.
Solution 2:
The phrase "sigh of contentment" is common enough in published English to support a continuous Google Ngram line graph for the period 1830–2008:
The frequency of the expression is not particularly high, but it has been greater than 0% every year since 1852. For many English speakers, it is virtually a set phrase and certainly is not widely interpreted as a "negative expression," contrary to the opinion attributed to the poster's teacher.
Her are the three earliest instances I found in the course of several database searches. First, from Laughton Osborn, "Epistle III. To Juvenal," in Arthur Carryl, a Novel (1841):
Thus have I seen the new mother, whose fountains were over replenish'd, / when from the rose-tinted tube her cherub, with sigh of contentment, / Closing its innocent eyes, had withdrawn its soft lips in repletion, / Yield, with a smile, the rich fount, to be press'd by the soft gums of a kitten; / ...
From a translation by Hugomont of Altarouche, "The Ounce of Snuff," in The Literary Garland (August 1843):
My uncle never takes a pinch of snuff (and he takes a good many) without heaving a sigh of contentment and satisfaction, glancing at the same time at the lid of his snuff-box, which is adorned with a highly finished portrait of a fair and handsome girl.
From Confessions of an Homœopathist (1846):
Pōble, who was impetuously struggling into his clothes, pointed to a large jug, into hich the Courier plunged, with an avidity and heartiness which soon gave him a view of the bottom, and then laid it down with a sigh of contentment, remarking, "Good—right good for Leipsic!"
From Caroline Gascoigne, Evelyn Harcourt: A Novel (1847):
"If you are pleased, I am indeed rewarded," replied Helen, with a sigh of contentment. "But now, Evelyn, dearest, let me hear you sing—let me hear if you still retain your wonderful talent. It is long since I have heard you, you know."
And from Eugène Sue, The Commander of Malta (1847):
On finding himself alone, Peyroü heaved a sigh of contentment, as if he at length found himself master of his little kingdom.
In short, a "sigh of contentment" is an expression of pleasure or satisfaction and is widely understood to be positive, not negative—as is the case in all five instances cited above. Although these examples are drawn from the 1840s, they use "sigh of contentment" in essentially the same way that people today use it.