Using 'frighten into doing' suggests a strong negative emotion inducing action. It's more intense than just persuading or convincing. The thunderstorm frightened the campers into packing up.
Pro Tip 2/3
Lack of Consent
'Frighten into doing' implies that the action is against the person's will and is not a choice made freely. The fierce dog frightened the jogger into changing his route.
Pro Tip 3/3
Power Dynamics
This phrase often suggests a power imbalance between the person frightening and the person being frightened. The boss frightened the employees into working overtime.