In my opinion, emotions are quite logical and a fundamental component of decision making. The assumptions we make and the preferences we have among otherwise equally valued options are rooted in emotion.
Let's take a look at anger, first of all. One responds in anger at a perceived injustice. Anger demands of us equity, the restoration of justice. A primitive impulse one may have when angered is to use aggression against the offending party; this is an impulse a psychopath, for example, has poor ability to control. A person who has good control of his or her emotions will not become especially angry, or even angry at all, when, for example, he or she is cut off while driving because the slight is too small for any action to be reasonable. An emotionally mature people, when insulted, will probably seek clarification instead of feeling indignation; perhaps they misunderstood. Anger, of course, is not an emotion that should simply be denied. A person might feel a sort of righteous indignation at the government's handling (or mishandling) of social injustices and will act on this form of anger constructively: perhaps by starting a petition or running for office to do things better. Most social justice movements are based on a mix of guilt, anger, compassion, and more that compels people to "do something about it." Another form of sublimination might be sports or even creative pursuits. The results of anger, properly applied, can be social improvement, self-improvement, maintained dignity, and creative output.
Sadness promotes attachment, strangely enough. To value one's attachment to someone or something or even a goal or ideal, one must know what it is like to have lost. In psychodynamic terms, the first emotional bond is with oneself; this is narcissism. At the egocentric state, one's own needs and desires are recognized. Emotional maturation expands the net of recognition to one's loved ones, family and friends, and then to the greater community and finally to humanity as a whole and even other forms of life. As one's love and compassion grow, more sophisticated mechanisms are needed to judge and assess one's own desires against the needs and rules of society, as well as a consideration of other individuals' situation. Narcissists fail to develop the empathy needed to incorporate others' needs into their thinking. Anyway, sadness results when something has been lost, usually by chance or forces beyond one's control, or through some failure to reach a goal or obtain/support an ideal. We grieve over a dead relative and miss the friend who has moved away. We feel guilty for having violated a personal ethic or a social more. We feel frustrated when we have repeatedly failed to reach our goals; this is where sadness blends into anger.
One might feel anger, sadness, fear, or some other emotions as a response to a certain event. Many factors are involved. If something was preventable and purposeful, we might feel anger. If something was an unavoidable circumstance that's part of life (e.g., death and illness), we might feel sad. If we feel a loss but also believe it was preventable, we may feel sad and angry at the same time. A person who has a highly assertive, even dominating, personality has a higher probability of feeling anger as the reaction, especially if they are narcissistic, with little if any emotional ties to others. One person might feel sad for the Iraqis, guilty to be a citizen of the country that invaded the Iraqis' country, angry at their government for not having deliberated and choosing a more peaceful option, and compassion for the situation the Iraqis are in; this person may realize that, despite the good Saddam Hussein's ouster can do, the disruption and threat to many thousands of Iraqi and American soldiers' lives may not be worth it when some other means could be found. Another person may have been treated fairly well all his life; he was popular in school and always followed the latest trends. He has come to identify himself with the established authorities and ways of doing things because it has served him well in the past. He is shocked that some people, his compatriots, might protest the war (and by implication the establishment that benefits him) and angry that the Iraqis are resisting the U.S.'s efforts to "help" that "pitiful" nation. He has rarely been disadvantaged in life and has found his naturally competitive temperament has led him to the success and happiness he desires; he reasons that the misfortune of others is their own fault because they did not compete hard enough to advance themselves, so he believes these less fortunate people should accept any help the successful throw at them graciously. He has had little reason to develop significant empathy for those much different from himself.
Happiness is another emotion that bonds us to others. When we feel happy around people, we grow to love them. Our understanding of them grows, as it does in sadness. When our loved ones become sad, we take on their sadness and are thus compelled to alleviate it to restore happiness.