The underneath portions of your arms are your tricep muscles. It is called the tricep muscle because there are three "heads" to the muscle. This muscle involves the extension of your elbow, so pushing yourself up and away from the arms of the chair you're sitting on works your tricep muscles. Many exercises strengthen your underarm muscles, but start with the basics.
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A tricep push-down is an exercise done on a cable-pulley system available at most gyms. There will be a small bar or a rope that you can attach to the cable-pulley. Raise the cable so the attachment of the bar or rope and the cable are at about eye level. The starting position of your hands will be at about the top of your chest. Face the cable and pull the bar or rope downward while keeping your elbows close to the side of your torso, to full elbow extension. Return upward until the attachment point is about eye level once more.
Pushups work your triceps as well as your chest, and the same goes for the bench press. To perform a pushup, place your hands on the ground by the side of your body so that your elbows are pointing behind you at roughly a 45-degree angle. Keep your back, shoulders and hips in line as you drop your chest between your hands. The goal is to touch the chest to, or close to, the ground, then return to the starting position. Always have your fingers facing forward.
If your strength doesn't support your weight yet, place your knees on the ground and slightly back so your chest can drop comfortably between your hands. To modify the intensity placed on the triceps, simply bring your hands closer together underneath your chest.
A dip primarily targets your tricep muscles and somewhat your pectoral muscles. You will need two parallel bars of the same height for this exercise. Many gyms have a place for dip bars, and some supply assisted dip machines. Grip around the bars so that your knuckles are facing out, and extend the elbows. Then take your feet off the ground. You are now in a structural lockout supporting your body weight if using an unassisted machine. Begin to bend your elbows -- without flaring them outward -- to lower yourself. Lower yourself until the elbow joint is at a 90-degree angle, then press upward. Try to keep your elbows close to your body throughout.
Sets, Repetitions and Rest Optimal for Strength Training
According to the National Strength and Conditioning Association, there are several sets of reps you should complete to achieve optimal strength results. A recommended rest period in between sets benefits strength training. You should attempt to complete 2 to 6 sets of 6 to 10 reps for strength training. There is a drop-off of significant gains made after completing three sets of an exercise, so 2 to 3 sets is best. The rest period for strength training should be ideally 2 to 5 minutes in between sets.