Final answer:
The case that established that police must provide warnings of Miranda rights to protect against self-incrimination under the Fifth Amendment is Miranda v. Arizona. This case requires procedural safeguards to ensure individuals are aware of their right to remain silent and to have counsel present during interrogations.
Step-by-step explanation:
The case that establishes that the standard does not involve subjective inquiries into the law enforcement officer's thoughts, feelings, and motivations is Illinois v. Gates, which established the "totality of circumstances" test for finding probable cause under the Fourth Amendment. However, the question seems to be inquiring about the Fifth Amendment protections against self-incrimination during police interrogations. Here, the landmark case is Miranda v. rizonAa, which determined that prosecutors cannot use statements from custodial interrogation of defendants unless procedural safeguards are used to secure the privilege against self-incrimination. These include warnings of the right to remain silent and the right to have counsel present during interrogations, which are now known as Miranda rights. The importance of Miranda v. Arizona cannot be overstated as it ensures that individuals are aware of their Fifth Amendment rights during the critical moments of custodial interrogation.