Mathematics

Problem

Consider the four standard products that involve

\sin x,\,\cos x,\,\sin y

and

\cos y

. a. (i) Write

\sin x\,\cos y+\cos x\,\sin y

as a single trigonometric function of a single angle. b. (ii) Write

\cos x\,\cos y-\sin x\,\sin y

as a single trigonometric function of a single angle. c. (iii) Give a brief geometrical or algebraic explanation (2–3 sentences) to show why the sign pattern in the product determines whether the compressed angle is

(x+y)

(x-y)

. d. (iv) Without further derivation, state the single-angle equivalents of

\sin x\,\cos y-\cos x\,\sin y

and of

\cos x\,\cos y+\sin x\,\sin y

. Explain in one sentence how your answer in part (iii) helps you decide the signs.

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