Two excellent answers so far. I'd add two things...
Firstly, I can't recommend reading "Quiet" by Susan Cain enough. She discusses the place of the introvert in different cultures and highlights areas where introverts can have a natural advantage over extroverts.
Secondly, I worked in a company that ran a Myers-Briggs training course. (If you aren't familiar with it, it classifies personality types as a combination of Introvert/Extrovert, Sensing/Intuition, Thinking/Feeling and Judging/Perceiving). I've forgotten most of it by now but I'll always remember the final exercise we did as it has helped me a lot. (For what it's worth, I'd always have been very skeptical about that kind of wishy-washy training)
We had all done our Myers-Briggs test to determine our personality type. The instructor got us to write our name on a piece of paper twice - once with each hand. She asked "how did it feel with your dominant hand". She got answers like "natural, simple, normal". Then she asked about the other hand: "weird, strange, awkward, clumsy, etc.".
Next, she asked if, with practice, you could get better with your non-dominant hand. Everybody agreed that, over time, we could.
Her point was this: We had been classified as Introvert or Extrovert, Thinking or Feeling etc.. and this was the natural modus operandi for each of us. But, she explained, we were all capable of being the other type of the two, it just feels uncomfortable and takes a little more effort to do it. And, if we made a conscious effort to push outside of our natural comfort zone, we would, over time, feel less uncomfortable doing it.
That doesn't help you so much in the short term but it has worked wonders for me and given time, and a little effort, it can for you too. For example, I'm quite shy and don't really celebrate birthdays because I hate being the center of attention, yet I've performed stand up comedy and have actually enjoyed giving presentations to over 300 people in work. It's taken a lot of effort to get to that point but I'm a lot more comfortable now in situations where I'd have given anything to be home with a coffee and a good book.
In the end, my answer to your question invites you to a different approach: try to improve your handling of situations where shyness would normally result in you being perceived as cold or arrogant, in order to help minimize that perception. Take it easy, small steps, and don't quit if you haven't succeeded from the first time(or second, or third).