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I recently learnt the formula $a \sinθ + b \cosθ = R \sin(θ + α)$ at school. How to prove it? Thanks.

raindrop
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Robin Ting
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2 Answers2

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HINT Suffices to show there exist $a,b,R$ such that $$a\sin t + b\cos t = r\sin(t+z).$$

Note that $$ \sin(t+z) = \sin t \cos z + \cos t \sin z $$ and pick $a/r = \cos z, b/r = \sin z$ where $a^2+b^2 = r^2$.

Steven Alexis Gregory
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gt6989b
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  • Probably $a,b$ are given and $R,\alpha$ should be found. If $a^2+b^2=0$ we may take $R=0$ whereas $\alpha$ can be anything. In the non-trivial case $R:=\sqrt{a^2+b^2}$ and $\alpha$ may be determined as $z$ from the above answer. – Jens Schwaiger Aug 31 '18 at 02:50
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Let $\dfrac{b}{a}=\tan\alpha$ ($a\neq0$) then $$LHS=a\left(\sin\theta+\dfrac{b}{a}\cos\theta\right)=a\left(\sin\theta+\tan\alpha\cos\theta\right)=\dfrac{a}{\cos\alpha}\left(\sin\theta\cos\alpha+\sin\alpha\cos\theta\right)=\dfrac{a}{\cos\alpha}\sin(\theta+\alpha)$$

Nosrati
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