two words: Body language.

Body language takes up 55% of our communication. Paralanguage (tones or how we say it) is another 38%. That only leaves 7% for actually words, so they're virtually useless when describing how a character is feeling.

If she is happy: "Her eyes were bright, and two high spots of color reddened her cheeks"

If he is angry: "His eyes were narrowed, and his fists were clenched into bloodless claws."

The above sentences don't have the actual emotion anywhere. All I used was body language.