Mathematics

Mathematics

Problem

Evaluate sin⁡75∘ cos⁡15∘−cos⁡75∘ sin⁡15∘\sin 75^\circ\,\cos 15^\circ-\cos 75^\circ\,\sin 15^\circ without using a calculator.
Intuitive explanation

This is a sine difference formula in disguise. Notice that you have the pattern sin⁡Acos⁡B−cos⁡Asin⁡B\sin A \cos B - \cos A \sin B, which is exactly sin⁡(A−B)\sin(A - B). Once you recognize this structure and apply the formula, the angles will subtract to give you something you already know.

Intuitive explanation

This is a sine difference formula in disguise. Notice that you have the pattern sin⁡Acos⁡B−cos⁡Asin⁡B\sin A \cos B - \cos A \sin B, which is exactly sin⁡(A−B)\sin(A - B). Once you recognize this structure and apply the formula, the angles will subtract to give you something you already know.

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