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How can I solve the following equation? $$-20=15 \sin \theta- 30.98 \cos \theta$$

I can't think of any way to solve it. You can't factor out cosine because of the annoying little negative twenty, and if you divide by cosine you also get nowhere.

Daniel R
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Jason
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    http://math.stackexchange.com/questions/194331/solution-to-equation-a-cdot-cos-theta-b-cdot-sin-theta-c – Ishfaaq Sep 13 '14 at 02:42
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    There's a method for this, and it's a bit involved. You begin by dividing both sides of the equation by $\sqrt{15^2 + 30.98^2}$. At that point you have turned it into an equation of the form $a \sin \theta + b \cos \theta = c$, where $a^2 + b^2 = 1$. Now since the point $(a,b)$ is on the unit circle, you can find its angular coordinate $\alpha$. Then your equation becomes $\cos \alpha \sin \theta + \sin \alpha \cos \theta = c$, or, equivalently, $\sin(\theta + \alpha) = c$. Next solve this equation to find $\theta + \alpha$, and finally $\theta$. – Dave Sep 13 '14 at 02:42

2 Answers2

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There are many approaches. One way is to bring the $\sin\theta$ stuff to one side, and the rest to the other. Square both sides, and replace $\sin^2\theta$ by $1-\cos^2\theta$. We get a quadratic in $\cos\theta$. Solve, and for each solution check whether it is a solution of the original equation.

For another approach, consider more generally $a\cos\theta+b\sin\theta=c$. Rewrite as $$\sqrt{a^2+b^2}\left(\frac{a}{\sqrt{a^2+b^2}}\cos\theta+\frac{b}{\sqrt{a^2+b^2}}\sin\theta\right)=c.$$

Find an angle $\varphi$ whose sine is $\frac{a}{\sqrt{a^2+b^2}}$ and whose cosine is $\frac{b}{\sqrt{a^2+b^2}}$. Then our equation says that $$\sin(\theta+\varphi)=\frac{c}{\sqrt{a^2+b^2}}.$$ Now find $\theta+\varphi$, and then $\theta$.

André Nicolas
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If you're familiar with basic vectors, you may reason the $a\cos\theta + b\sin\theta $ formula like this $$20 = 30.98 \cos \theta - 15\sin \theta = \langle 30.98, -15 \rangle \bullet \langle \cos \theta, \sin\theta \rangle = C \cos(\theta - \alpha) $$

$C = \sqrt{30.98^2 + 15^2}$
$\alpha = \arctan\left(- \dfrac{30.98}{15}\right)$

AgentS
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