Quick questions on Auxiliary angle: writing $a \sin x + b \cos x$ as $R \sin(x + \alpha)$

2short Q&A pairs drawn directly from our worked dot-point answer. For full context and worked exam questions, read the parent dot-point page.

What is wrong quadrant for

\alpha

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\tan \alpha = \frac{b}{a}

has two candidate angles per period. Choose the one consistent with the signs of

R \cos \alpha = a

and

R \sin \alpha = b

What is negative

R

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The convention is

R > 0

. If you ever write

-R \sin(x + \alpha)

, fold the minus into the phase via

-R\sin(x+\alpha) = R\sin(x+\alpha+\pi)

to keep

R

positive.

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