Trigonometric equations

A trigonometric equation is an equation involving trigonometric functions of an unknown angle. Because these functions are periodic, every such equation has infinitely many solutions, and we must learn to write all of them in one expression , the general solution.

Principal value vs general solution

If $\sin x = \dfrac{1}{2}$, then $x = \dfrac{\pi}{6}$ works. But so does $\dfrac{5\pi}{6}$ (the same height on the other side of the unit circle), and also $\dfrac{\pi}{6} + 2\pi$, $\dfrac{5\pi}{6} + 2\pi$, $\dfrac{\pi}{6} - 2\pi$, and so on. The general solution captures every one of these.

The three master rules

For each basic equation we write the general solution in closed form.

$\sin x = \sin\alpha$

$x = n\pi + (-1)^n \alpha, \quad n \in \mathbb{Z}.$

Verify at $n = 0$: $x = \alpha$. At $n = 1$: $x = \pi - \alpha$. At $n = 2$: $x = 2\pi + \alpha$. Etc.

$\cos x = \cos\alpha$

$x = 2n\pi \pm \alpha, \quad n \in \mathbb{Z}.$

$\tan x = \tan\alpha$

$x = n\pi + \alpha, \quad n \in \mathbb{Z}.$

(Period of $\tan$ is $\pi$, not $2\pi$.)

Three special cases

Strategy for general equations

1. Reduce to a basic form. Use identities to get $\sin = \sin$, $\cos = \cos$, or $\tan = \tan$.
2. Square only when necessary. Squaring can introduce extraneous solutions; check at the end.
3. Factor. Quadratics in $\sin x$ or $\cos x$ factor and split into two equations.
4. Write the general solution. Apply the master rules and combine.

Solving $a \sin x + b \cos x = c$

Convert to $R\sin(x + \phi) = c$ with $R = \sqrt{a^2 + b^2}$. The equation has solutions iff $|c| \le R$. If yes, $\sin(x + \phi) = c/R$, and we solve as $\sin\alpha = c/R$.

Worked examples

Example 1. Solve $\sin x = \dfrac{\sqrt{3}}{2}$.

$\sin x = \sin\dfrac{\pi}{3}$, so $x = n\pi + (-1)^n \dfrac{\pi}{3}$, $n \in \mathbb{Z}$.

Example 2. Solve $2\cos^2 x - 1 = 0$.

$\cos^2 x = \dfrac{1}{2}$, so $\cos x = \pm \dfrac{1}{\sqrt{2}}$. The two cases:

• $\cos x = \dfrac{1}{\sqrt{2}} \Rightarrow x = 2n\pi \pm \dfrac{\pi}{4}$.
• $\cos x = -\dfrac{1}{\sqrt{2}} \Rightarrow x = 2n\pi \pm \dfrac{3\pi}{4}$.

Compactly: $x = n\pi \pm \dfrac{\pi}{4}$, $n \in \mathbb{Z}$. (Or use $\cos 2x = 0$: $2x = (2n+1)\pi/2$, $x = (2n+1)\pi/4$.)

Example 3. Solve $\tan x = -1$.

$\tan x = \tan\!\left(-\dfrac{\pi}{4}\right)$, so $x = n\pi - \dfrac{\pi}{4}$.

Example 4. Solve $\sin x + \cos x = 1$.

Convert: $\sqrt{2}\sin\!\left(x + \dfrac{\pi}{4}\right) = 1$, so $\sin\!\left(x + \dfrac{\pi}{4}\right) = \dfrac{1}{\sqrt{2}} = \sin\dfrac{\pi}{4}$.

So $x + \dfrac{\pi}{4} = n\pi + (-1)^n \dfrac{\pi}{4}$. Splitting by parity:

• $n$ even: $x + \dfrac{\pi}{4} = 2k\pi + \dfrac{\pi}{4}$, so $x = 2k\pi$.
• $n$ odd: $x + \dfrac{\pi}{4} = (2k+1)\pi - \dfrac{\pi}{4}$, so $x = 2k\pi + \dfrac{\pi}{2}$.

Combined: $x = 2k\pi$ or $x = 2k\pi + \dfrac{\pi}{2}$ for $k \in \mathbb{Z}$.

Example 5 (harder). Solve $\sin 5x = \sin 3x$.

Bring to one side and use sum-to-product: $\sin 5x - \sin 3x = 2 \cos 4x \sin x = 0.$ So $\cos 4x = 0$ or $\sin x = 0$.

• $\cos 4x = 0 \Rightarrow 4x = (2n+1)\dfrac{\pi}{2} \Rightarrow x = (2n+1)\dfrac{\pi}{8}$.
• $\sin x = 0 \Rightarrow x = m\pi$.

General solution: $x = m\pi$ or $x = (2n+1)\dfrac{\pi}{8}$.

Try it yourself

1. Solve $\cos x = \dfrac{1}{2}$.
2. Solve $\tan x = \sqrt{3}$.
3. Solve $\sin 2x = \sin x$.
4. Solve $\cos 3x = \cos 2x$.
5. Solve $2\sin^2 x - 3\sin x + 1 = 0$.
6. Solve $\sin x + \sin 3x = 0$.
7. Solve $\sqrt{3}\sin x + \cos x = 1$.
8. Solve $\tan x + \cot x = 2$.
9. Solve $\sin^2 x - \cos^2 x = \dfrac{1}{2}$ (use $\cos 2x = \cos^2 - \sin^2$).
10. Find all $x \in [0, 2\pi]$ with $\sin x = \cos x$.
11. Solve $\cos x \cos 2x = \dfrac{1}{4}$. (Use product-to-sum.)
12. Solve $\sin x + \sin 2x + \sin 3x = 0$.

Pitfalls / Tricks

• For $\sin x = 0$, the general solution is $x = n\pi$, not $x = 2n\pi$ , the period of $\sin x$ is $2\pi$, but zeros occur every $\pi$.
• When you square both sides, check the final answers , squaring can introduce spurious roots.
• $\tan x$ has solutions $x = n\pi + \alpha$, without the $(-1)^n$ factor (which is only for $\sin$).
• Insight. Almost every trigonometric equation reduces to $\sin f(x) = \sin g(x)$ or $\cos f(x) = \cos g(x)$. Master these two patterns and you can solve nearly any problem.