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Algebra Difficulty 2.0 Junior Prove it Canada

If $f(t)=\sin\left(\pi t-\frac{\pi}{2}\right)$ , what is the smallest positive value of $t$ at which $f(t)$ attains its minimum value?

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project. Solutions are the publisher's, linked not copied.