I'm fluent in Japanese, but I've just started learning Korean, it's fun finding connections. I actually found this forum because I wondered if Korean 다라 (tari, meaning leg) and Japanese 足りる (tariru, meaning to be enough, but it uses the kanji for leg) were related and found a post about that. The majority of similar words are Sinitic and there are quite a lot of them. Some words are pretty obvious, like 주의 and 注意 (both chuui) others are not similar enough to be obvious without context or a dictionary, like 확인 and 確認 (hwagin and kakunin). I'm curious if there are any common patterns besides the ones I've noticed.
っ and つ (small tsu and tsu) generally become ㄹ (l)
Examples: 특별/特別 (teukbyeol / tokubetsu) 벌표/発表 (beolpyo / happyou) 설치/設置 (seolchi / secchi)
く (ku) generally stays a k sound, becoming ㄱ (k)
Examples: 세탁洗濯 (setak / sentaku) 입력/入力 (ibryeok / nyuuryoku)
long o and long e おう and えい often becomes ㅇ (ng)
Examples: 중학교/中学校 (chunghakkyo / chuugakkou) 동부/東部 (dongbu /toubu) 설명/説明 (seolmyeong / setsumei) 선생/先生 (seonsaeng / sensei)
Let me know if you know any other general patterns that might be helpful to know. ^^