How do I rewrite 2 cos x + 4 sin x as one sin function?
This question makes use of the sin addition formula. It may be stated as sin (A + B) = sinA cosB + sinB cosA . We want to rewrite 2 cosx + 4 sinx in the form R sin (x + a), so firstly work out what R sin(x +a) is, expanded. By using the formula above, we get R sin(x + a) = Rinxcosa + Rsinacosx or (R cos a) sinx + (R sin a) cosx, where the parts in the brackets are the constants.
We can therefore equate the constants to the constants given in the original expression, i.e. 4 and 2, so we get that R cos a = 4 and R sin a = 2. Making use of two more trig formulae, we can work out what R and a are. For example. cos^2 x + sin^2 x = 1, meaning R = sqrt (R cos^2 a + R sin^2 a). To work out a, use tan a = sin a / cos a.
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