Triune means three in unity. When it applies to God it pertains to three persons in one Godhead. The word trinity is not found in the Bible and the Old Testament does not directly teach it. However in Genesis God said, "'Let Us make man in Our Image, according to Our likeness'" (Ge 1:26). Then after man was created he said, "'Behold, the man has become like one of Us, knowing good and evil'" (3:22). The Trinity is inferred by the use of the plural pronoun "us." Later God said, "'Come, let Us go down and there confuse their language so that they will not understand one another's speech'" (11:7). Then Isaiah "heard the voice of the Lord, saying, 'Whom shall I send, and who will go for Us?'" (Isa 6:8).
The three, equal members of the Trinity cooperate because they are one God. But tritheists believe that the three members of the Godhead are entirely separate and have individual hierarchical importance. Also, those who believe in Sabellianism say God takes the form called for at the time saying that there is only one God and no multiple personages. However, the Bible has passages which mention A) the Father and Jesus, B) the Father and Spirit, C) the Son and Spirit, and D) God, Holy Spirit and Christ. In the first case "In the beginning was the Word, and the Word was with God, and the Word was God" (Jn 1:1). "Who has established all the ends of the earth? What is His name or His Son's name?" (Pr 30:4). "I will surely tell of the decree of the Lord: He said to Me, 'You are My Son, today I have begotten You'" (Ps 2:7). Because "God so loved the world" (Jn 3:16) he "sent forth His Son" (Gal 4:4). Jesus says, "'For I have come down from heaven, not to do My own will, but the will of Him who sent Me'" (Jn 6:38). Their relationship is such that "'no one knows the Son except the Father; nor does anyone know the Father except the Son'" (Mt 11:27). Then because of what was accomplished at the resurrection "whoever confesses that Jesus is the Son of God, God abides in him, and he in God" (1Jn 4:15). "It was the Father's good pleasure for all the fullness to dwell in Him" (Col 2:9). John said "our fellowship is with the Father, and with His Son Jesus Christ" (1Jn 1:3).
In the second case "the thoughts of God no one knows except the Spirit of God" (1Co 2:11). It is "'Not by might nor by power, but by My Spirit,' says the Lord of hosts" (Zec 4:6). The interrelationship is seen as when "the Lord came down in the cloud and spoke to him; and He took of the Spirit who was upon him and placed Him upon the seventy elders" (Nu 11:25). Nehemiah relates to the Father and Spirit by saying "You bore with them for many years, and admonished them by Your Spirit through your prophets" (Neh 9:30). "He who rejects this is not rejecting man but the God who gives His Holy Spirit to you" (1Th 4:8). In the third case Jesus "was declared the Son of God with power by the resurrection from the dead according to the Spirit" (Ro 1:4). Isaiah cites the Messiah and Spirit with "for He said, 'Surely they are My people but they rebelled and grieved His Holy Spirit'" (Isa 63:10). A unilateral example is when "the Spirit of the Lord came mightily upon David from that day forward" (1Sa 16:13). The Holy Spirit "intercedes for the saints according to the will of God" (Ro 8:27).