How might monarchs react to the ideas of the Enlightenment?

Some embraced them, some were indifferent, some rejected them. You’re speaking of an era that spanned decades during which there were scores of not hundreds of reigning monarchs, Princes, Dukes, and so on just in Europe alone, not to mention beyond European shores. You’re asking too broad and too general a question to answer with any specificity. You might want to refine it to a specific or countries, or a more specific point in time, or how they reacted to it as a challenge to something specific like the divine right of kings.

u.s.a

china

india

and

japan

roosevelt believed that in light of the country's recent military successes, it was unnecessary to use force to achieve foreign policy goals, so long as the military could threaten force. this belief, and his strategy of “speaking softly and carrying a big stick,” shaped much of roosevelt's foreign policy.