# Solving $\sin x + \sqrt 3 \cos x = 1$ - is my solution correct?

I have an equation that I'm trying to solve:

$$\sin x + \sqrt 3 \cos x = 1$$

After pondering for a while and trying different things out, this chain of steps is what I ended up with:

$$\sin x + \sqrt 3 \cos x = 1$$

$$\sin x = 1 - \sqrt 3 \cos x$$

$$\left(\sin x \right)^2 = \left(1- \sqrt 3 \cos x\right)^2$$

$$\sin^2 x = 1 - 2 \sqrt 3 \cos x + 3 \cos^2 x$$

$$2 \sqrt 3 \cos x - 3 \cos^2 x = 1 - \sin^2 x$$

$$2 \sqrt 3 \cos x - 3 \cos^2 x = \cos^2 x$$

$$2 \sqrt 3 \cos x = \cos^2 x + 3 \cos^2 x$$

$$4 \cos^2 x = 2 \sqrt 3 \cos x$$

$$\frac{4 \cos^2 x}{\cos x} = 2 \sqrt 3$$

$$4 \cos x = 2 \sqrt 3$$

$$\cos x = \frac{2 \sqrt 3}{4}$$

$$\cos x = \frac{\sqrt 3}{2}$$

The fraction $\frac{\sqrt 3}{2}$ can be rewritten as $\cos \left(\pm \frac{\pi}{6}\right)$, so my solutions are:

$$\cos x = \cos \left(\frac{\pi}{6}\right) \quad \text{or} \quad \cos x = \cos \left(-\frac{\pi}{6}\right)$$

$$x = \frac{\pi}{6} + 2\pi n \quad \text{or} \quad x = -\frac{\pi}{6} + 2\pi n$$

Since I earlier on exponentiated both sides I have to check my solutions:

$$x = \frac{\pi}{6} + 2\pi \Rightarrow \text{LHS} = \sin \left(\frac{\pi}{6} + 2\pi\right) + \sqrt 3 \cos \left(\frac{\pi}{6} + 2\pi\right) = 2 \not = \text{RHS}$$

$$x = -\frac{\pi}{6} + 2\pi \Rightarrow \text{LHS} = \sin \left(-\frac{\pi}{6} + 2\pi\right) + \sqrt 3 \cos \left(-\frac{\pi}{6} + 2\pi\right) = 1 = \text{RHS}$$

Leaving $x = -\frac{\pi}{6} + 2\pi n$ as the answer since its positive counterpart was not equal to $1$.

$$\text{Answer:} \: x = -\frac{\pi}{6} + 2\pi n$$

Have I done anything wrong or does this look good? I haven't really done this before so I feel uncertain not just about the solution, but also my steps and notation...

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Hint: if you substitute the result of x for n = 0 , 1, 2, ..., are both sides of the equation equal to 1? You also lost a solution, but I'd like to see if you can find it. Lastly, third line got a little messed up on what is displayed. –  Amzoti Sep 23 '12 at 22:39
You lost the solutions with $\cos x=0,\ \sin x=1$ when you divided by $\cos x$. –  kiwi Sep 23 '12 at 22:39
You can also compare with Wolfram Alpha: "wolframalpha.com/input/?i=Solve%5BSin%5Bx%5D%2B%20Sqrt%5B3%5D*Cos%5B‌​x%5D%20%3D%3D%201%2C%20x%5D&t=crmtb01" –  Amzoti Sep 23 '12 at 22:46

You went a little bit astray after $4 \cos^2 x = 2 \sqrt 3 \cos x$, when you divided by $\cos x$: what if $\cos x=0$?

It’s better at that point to bring everything to one side and factor: $4\cos^2x-2\sqrt3\cos x=0$, so $2\cos x(2\cos x-\sqrt3)=0$. Now appeal to the fact that if a product is $0$, at least one of the factors must be $0$. Obviously $2\ne 0$, so either $\cos x=0$, or $2\cos x-\sqrt3=0$. As it happens, both of these possibilities give you solutions. You found the second set, but not the first set.

If $\cos x=0$, we need $\sin x=1$ to have a solution. If $\sin x=1$, $\cos x$ is automatically $0$, so you just need to find the solutions to $\sin x=1$ to complete your solution.

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There is a standard method for solving equations of the form:

$$A \sin x + B \cos x = C$$

Divide both sides by $\sqrt{A^2 + B^2}$:

$$\frac{A}{\sqrt{A^2 + B^2}} \sin x + \frac{B}{\sqrt{A^2 + B^2}} \cos x = \frac{C}{\sqrt{A^2 + B^2}}$$

Find $\theta \in [0, 2\pi)$ so that:

$$\sin \theta = \frac{B}{\sqrt{A^2 + B^2}} \\ \cos \theta = \frac{A}{\sqrt{A^2 + B^2}} \\$$

And $\phi \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ so that:

$$\sin \phi = \frac{C}{\sqrt{A^2 + B^2}}$$

If you cannot find such a $\phi$, then the equation doesn't have any solutions. (For example, if $\frac{C}{\sqrt{A^2 + B^2}} > 1$.

Thus:

$$\cos \theta \sin x + \sin \theta \cos x = \sin \phi$$

Using the angle sum identity, we have:

$$\sin(x + \theta) = \sin \phi$$

Therefore:

\begin{align*} x_1 &= \phi - \theta + 2 \pi n \\ x_2 &= \pi - \phi - \theta + 2 \pi n \end{align*} (Where $n \in \mathbb{Z}$)

Now, let's apply this method to your question. We have:

$$A = 1, B = \sqrt{3}, C = 1 \\ \sqrt{A^2 + B^2} = 2 \\ \sin \theta = \frac{\sqrt{3}}{2}, \cos \theta = \frac{1}{2}, \theta = \frac{\pi}{3} \\ \sin \phi = \frac{1}{2}, \phi = \frac{\pi}{6}$$

Thus:

$$x_1 = -\frac{\pi}{6} + 2 \pi n \\ x_2 = \frac{\pi}{2} + 2 \pi n \\$$

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You can collapse the left-hand side into a single sine function: $$\sin(x)+\sqrt3\cos(x) = 2\sin(x+\pi/3)$$ Then, dividing by two, all that remains is to solve the following: $$\sin(x+\pi/3) = \frac{1}{2}$$

Wikipedia has an article on useful trigonometric identities, including linear combinations of sin and cos.

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There's a nice trick: $$\sin x + \sqrt 3 \cos x = 1 \\\\ = 2 \left(\frac{1}{2}\sin x + \frac{\sqrt 3}{2} \cos x\right) \\\\ = 2\left(\cos\left(\frac{\pi}{3} + 2k\pi\right)\sin x + \sin\left(\frac{\pi}{3} + 2k\pi\right)\cos x \right)\\\\= 2\sin\left(x + \frac{\pi}{3} + 2k\pi\right) = 1.$$

When is $$\sin\left(x + \frac{\pi}{3} + 2k\pi\right) = 1/2$$

true?

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Note that the equation $\sin x + \sqrt 3 \cos x = 1$ is not equivalent to the equation $(\sin x)^2=(1−\sqrt{3} \cos x)^2$.

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