Consider the expression .

MATHEMATICS · Multi-part · A Level · Mathematics · Trigonometry · Trigonometry

Problem

Consider the expression sin⁡αcos⁡β+cos⁡αsin⁡β\sin\alpha\cos\beta+\cos\alpha\sin\beta. (i) State the identity that relates sin⁡(α+β)\sin(\alpha+\beta) to the product terms sin⁡αcos⁡β\sin\alpha\cos\beta and cos⁡αsin⁡β\cos\alpha\sin\beta. (ii) Without performing any algebraic manipulation, explain why sin⁡αcos⁡β+cos⁡αsin⁡β\sin\alpha\cos\beta+\cos\alpha\sin\beta can always be expressed as a single sine term. (iii) Decide whether the numerical value of sin⁡35∘cos⁡55∘+cos⁡35∘sin⁡55∘\sin35^\circ\cos55^\circ+\cos35^\circ\sin55^\circ equals sin⁡20∘\sin20^\circ or sin⁡90∘\sin90^\circ, giving a clear reason for your choice. (iv) Verify your conclusion by evaluating sin⁡35∘cos⁡55∘+cos⁡35∘sin⁡55∘\sin35^\circ\cos55^\circ+\cos35^\circ\sin55^\circ correct to four decimal places.

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