Is the takbir for bowing or prostrating sufficient when joining the congregation during them, or is a takbirat al-ihram required before it?

If a latecomer arrives while the Imam is bowing, he must say the opening Takbeer (Takbeerat al-Ihram), then say the Takbeer for bowing if he is able to do so. Or, the opening Takbeer suffices, and he then bows with the Imam, and that rak'ah counts for him. However, if he arrives while the Imam is prostrating, he says the opening Takbeer while standing, then prostrates. This rak'ah does not count for him due to missing its bowing and standing.