Trig question sum of sine and cosine functions (1 Viewer)

Jackson94 · Jan 21, 2012

Find R and a exactly
Rsina = 3 Rcosa = 3

Carrotsticks · Jan 21, 2012

$\begin{align*} R \sin \alpha &= 3 \qquad (1) \\ R \cos \alpha &= 3 \qquad (2) \end{align*} \\\\ $Solve simultaneously$ \quad \frac{(1)}{(2)}: \\\\ \frac{R \sin \alpha}{R \cos \alpha} = 1 \\\\ \tan \alpha = 1 \Rightarrow \alpha = 45^{\circ} $ or $ 225^{\circ} \\\\ $Substitute $ 45^{\circ} $ back to (1): $ \\\\ R \sin 45^{\circ} = 3 \\\\ R \times \frac{1}{\sqrt{2}} = 3 \Rightarrow R = 3 \sqrt{2} \\\\ $Substitute $ 225^{\circ} $ back to (1): $ \\\\ R \sin 225^{\circ} = 3 \\\\ R \times - \frac{1}{\sqrt{2}} = 3 \Rightarrow R = - 3 \sqrt{2} \\\\ $Therefore $ R = \pm 3 \sqrt{2} $ and $ \alpha = 45^{\circ} $ or $ 225^{\circ}$

However, I am guessing that you are using this question as a step for doing Auxiliary Angle method questions, so we would take the positive case for R.

$R = 3 \sqrt{2}$

AAEldar · Jan 21, 2012

You can also note that for R to be multiplied by something to get 3, then $\cos\alpha$ and $\sin\alpha$ must equal the same thing and this happens when $\alpha=45^\circ$ .

Carrotsticks · Jan 21, 2012

Personally, I do not like this method for solving Auxiliary Angle formulae. I very much prefer:

$R=\sqrt {(a^{2} + b^{2})} $ where a and b are the coefficients of sin and cos (or other way around, addition is commutative)$

EDIT: Referring to OP's method, which is often taught in schools. AAEldar's method by observation is perfectly fine.

Trig question sum of sine and cosine functions (1 Viewer)

Jackson94

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Carrotsticks

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AAEldar

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Carrotsticks

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