Remembering Allāh and declaring His greatness during Dhū al-Ḥijjah’s first ten days is a well established Sunnah. The Prophet (ﷺ) said: “There are no days that are greater before Allāh nor in which good deeds are more beloved to Him than these ten days, so recite the taḥlīl [i.e. saying Lā ilāha illā-Allāh], the takbīr [i.e. saying Allāhu Akbar], and the taḥmīd [i.e. saying Alhamdulillāh] frequently.” (Reported by Imām Aḥmad with this wording, and Shaykh Aḥmad Shākir authenticated it.)
Most scholars say this is the meaning of Allāh’s statement:
وَيَذْكُرُوا اسْمَ اللَّهِ فِي أَيَّامٍ مَّعْلُومَاتٍ
“Mention Allāh’s name during the known days.”
Therefore, remembering Allāh and proclaiming His greatness during Dhū al-Ḥijjah’s first ten days is legislated.
Remembering Allāh and declaring His greatness during the days of tashrīq is also legislated. Allāh said about the days of tashrīq, the three days after ʿĪd al-Aḍḥá:
وَاذْكُرُوا اللَّهَ فِي أَيَّامٍ مَّعْدُودَاتٍ
Remember Allāh during the appointed days.”
Furthermore, the Messenger (ﷺ) said: “Indeed, these days are for eating, drinking, and remembering Allāh.” (Reported by Abū Dāwūd  and al-Nasāʾī , and Shaykh al-Albānī authenticated it.)
Like the takbīr made during Dhū al-Ḥijjah’s first ten days, the takbīr made during the days of tashrīq is also vocalised.
Imām al-Bukhārī (رحمه الله) reported: “ʿUmar (رضي الله عنه) would make the takbīr in his tent when he was in Miná so much so that the people in the masjid would hear it. They would then follow suit. The merchants would also make the takbīr until Miná would tremble from the takbīr.” (Ṣaḥīḥ al-Bukhārī, vol 2, pg 20)
The takbīr made during the days of tashrīq is of two types. The general takbīr which is vocalised at any time. The other type is the specific takbīr, and it is vocalised immediately after the five daily prayers.
Imām al-Bukhārī (رحمه الله) reported: “Ibn ʿUmar (رضي الله عنهما) would make the takbīr in Miná after the prayers, in his bed, in his tent, in his gatherings, and while walking.” (Ṣaḥīḥ al-Bukhārī, vol 2, pg 20)
Therefore, we should also glorify Allāh’s name audibly and remember Him frequently during the days of tashrīq; in fact, we were encouraged to nourish ourselves during this time in order to fulfil this purpose.
Ibn Rajab (رحمه الله) said: “The Prophet’s (ﷺ) statement: ‘These days are for eating, drinking, and remembering Allāh’ suggests that we eat and drink to remember Allāh and to worship him during this time.” (Latāʾif al-Maʿārif, pg. 291)