Not to mention necessary to solve a lot of intermediate steps in Further (intermediate, advanced and research) Mathematics.
Maybe it's possible to learn about sines and cosines from Euler's identity up and forget about triangles altogether. This would enable you to do calculus, statistics, etc. But there's some value to teaching the theory of triangles as a beginner's introduction to pure mathematics (and hence the beauty in mathematics).
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What follows has to do with triangles, but is mostly offtopic.
One of the clearest memories of my childhood, somewhere between ages 8 and 10, is a German-made (but dubbed into my language) TV educational show that had men in togas walking around Classical Greek ruins and dramatizing some scenes -- Diogenes and Alexander and so on. But what has sticked in my mind and still burns red hot was Pythagoras and his mates, crouched and drawing lines in the sand. After Pythagoras explains his hypothenuse theorem, someone interjects:
"But this holds for all triangles?"
"All triangles. All triangles that have ever been drawn -- and all triangles that will never be drawn."
A sufficiently powerful brain scanning device probably can find this line somewhere inside my skull. It got me in big trouble: I'm not particularly intelligent, let alone disciplined, but I'm slogging through a Masters in mathematics, maybe dreaming about a PhD in physics because... I don't know. It's like when ghetto kids see some violent scene right in front of them and 20 years later their criminals. It sticks with you.