1a. She does not know to play the piano.
1b. She does not know how to play the piano.
1c. She does not know whether to play the piano.
2a. I do not know to cry or smile.
2b. I do not know how to cry or smile.
2c. I do not know whether to cry or smile.
Please correct the above.
Thanks.
1a. She does not know to play the piano.
… this would mean she is unaware that there is a need for the piano to be played. It probably does not have the meaning you are trying to give it.
1b. She does not know how to play the piano.
… correct.
1c. She does not know whether to play the piano.
… correct. This time the meaning is she is unsure whether she should play (at that moment) or not, but she does know how to play it.
If you had said,
She does not know whether to play the piano or the violin.
… the meaning could be as above or it could be that she is unsure which she wishes to learn. She cannot yet play either instrument.
2a. I do not know to cry or smile.
… this probably does not have your intended meaning. It indicates that the person is unaware that there is a need to cry or smile.
2b. I do not know how to cry or smile.
… this indicates that the person has never learned how to cry or smile. As this is highly unlikely, again I suspect it does not carry the meaning you want.
2c. I do not know whether to cry or smile.
… correct. The speaker is unsure which of the two reactions to display in response to whatever has just happened.