What Is a Trigonometric Equation?
A trigonometric equation is an equation that contains one or more trigonometric ratios of an unknown angle, such as $\sin x = \tfrac{1}{2}$, $2\cos^2 x - 1 = 0$, or $\tan x = \sqrt{3}$. Solving it means finding every value of the angle that makes the equation true.
Two kinds of answers exist, and keeping them apart is most of the battle:
Principal solution. The solution(s) that lie in the base interval $[0, 2\pi)$ (or $[0°, 360°)$). There are usually one or two.
General solution. Every solution, written in terms of an integer $n$ so the periodic copies are all included.
A trigonometric ratio is just the output of a function like sine on a given angle; if those ratios are unfamiliar, the trigonometric ratios reference lays them out before we use them here.
The General-Solution Formulas
These three results are the spine of the whole topic. In each, $n$ is any integer ($n \in \mathbb{Z}$).
$$\sin x = \sin \alpha ;\Rightarrow; x = n\pi + (-1)^n,\alpha$$
$$\cos x = \cos \alpha ;\Rightarrow; x = 2n\pi \pm \alpha$$
$$\tan x = \tan \alpha ;\Rightarrow; x = n\pi + \alpha$$
Why do the three differ? It comes down to where each function repeats its value.
Sine repeats every $2\pi$ but is also symmetric about $x = \tfrac{\pi}{2}$, so the $(-1)^n$ flips between the two solutions per cycle.
Cosine is symmetric about the $x$-axis, so each value comes from a $+\alpha$ and a $-\alpha$ branch, hence the $\pm$.
Tangent has the shortest period, $\pi$, so its solutions are spaced a single $\pi$ apart with no sign flip. Tangent's tighter spacing is a direct echo of its graph repeating twice as fast as the others.
Equation | General solution | Period feeding it |
|---|---|---|
$\sin x = \sin\alpha$ | $x = n\pi + (-1)^n \alpha$ | $2\pi$ |
$\cos x = \cos\alpha$ | $x = 2n\pi \pm \alpha$ | $2\pi$ |
$\tan x = \tan\alpha$ | $x = n\pi + \alpha$ | $\pi$ |
When the equation hands you a value that is not a standard angle, you reach for the inverse trigonometric functions to name $\alpha$ first, then drop it into the formula above.
How Do You Solve a Trigonometric Equation Step by Step?
The reliable routine is the same every time:
Isolate the trig ratio so the equation reads $\sin x = k$ (or $\cos$, $\tan$).
Find one known angle $\alpha$ whose ratio equals $k$ — a special angle if possible, otherwise an inverse-function value.
Apply the matching general-solution formula.
List principal solutions by setting $n = 0, 1$ if the question asks for the base interval.
Examples of Trigonometric Equations
Example 1
Find the principal solutions of $\sin x = \dfrac{\sqrt{3}}{2}$.
$\sin \dfrac{\pi}{3} = \dfrac{\sqrt{3}}{2}$, and sine is also positive in the second quadrant.
So $x = \dfrac{\pi}{3}$ and $x = \pi - \dfrac{\pi}{3} = \dfrac{2\pi}{3}$.
Final answer: $x = \dfrac{\pi}{3}, \dfrac{2\pi}{3}$.
Example 2
Solve $\cos x = \cos x - \sin x$ by first instinct, then correctly. Take $2\sin x = \sqrt{2}$.
The tempting move with $\sin x = \dfrac{1}{\sqrt{2}}$ is to write only $x = \dfrac{\pi}{4}$ and stop.
Test it against the graph: sine equals $\dfrac{1}{\sqrt{2}}$ at $\dfrac{\pi}{4}$, true — but also at $\dfrac{3\pi}{4}$, and again one full turn later. Stopping at one angle throws away every other solution, which is wrong the moment the question says "general solution."
The rescue is the formula. With $\sin x = \sin\dfrac{\pi}{4}$:
$$x = n\pi + (-1)^n \dfrac{\pi}{4}, \quad n \in \mathbb{Z}$$
Final answer: $x = n\pi + (-1)^n \dfrac{\pi}{4}$.
Example 3
Find the general solution of $\cos x = -\dfrac{1}{2}$.
$\cos \dfrac{2\pi}{3} = -\dfrac{1}{2}$, so $\alpha = \dfrac{2\pi}{3}$.
$$x = 2n\pi \pm \dfrac{2\pi}{3}, \quad n \in \mathbb{Z}$$
Final answer: $x = 2n\pi \pm \dfrac{2\pi}{3}$.
Example 4
Solve $\tan x = -\dfrac{1}{\sqrt{3}}$.
Tangent is negative in the second and fourth quadrants. A reference value is $\tan\dfrac{\pi}{6} = \dfrac{1}{\sqrt{3}}$, so $\alpha = -\dfrac{\pi}{6}$.
$$x = n\pi - \dfrac{\pi}{6}, \quad n \in \mathbb{Z}$$
Final answer: $x = n\pi - \dfrac{\pi}{6}$.
Example 5
Solve $2\cos^2 x + \cos x - 1 = 0$.
Treat it as a quadratic in $\cos x$. Let $c = \cos x$:
$$2c^2 + c - 1 = 0$$ $$(2c - 1)(c + 1) = 0$$ $$c = \dfrac{1}{2} \quad \text{or} \quad c = -1$$
For $\cos x = \dfrac{1}{2}$: $x = 2n\pi \pm \dfrac{\pi}{3}$.
For $\cos x = -1$: $x = (2n+1)\pi$.
Final answer: $x = 2n\pi \pm \dfrac{\pi}{3}$ or $x = (2n+1)\pi$.
Example 6
Solve $\tan 3x = \cot(x - 50°)$ for the smallest positive $x$ in degrees.
Rewrite $\cot\theta = \tan(90° - \theta)$:
$$\tan 3x = \tan\big(90° - (x - 50°)\big) = \tan(140° - x)$$
Using $\tan A = \tan B \Rightarrow A = n\cdot 180° + B$:
$$3x = 180°n + 140° - x$$ $$4x = 180°n + 140°$$ $$x = 45°n + 35°$$
The smallest positive value is at $n = 0$: $x = 35°$. Cofunction swaps like this lean on the cofunction identities connecting tangent and cotangent.
Final answer: $x = 35°$.
The most common first-instinct error here is keeping the equation in mixed $\tan$ and $\cot$ form and trying to equate angles directly — the cofunction conversion has to happen before any angle comparison, or the quadrants come out wrong.
Why Solving These Equations Matters - "The signal hits a target value"
The reason trigonometric equations exist is to answer a question every oscillating system asks: when does the wave reach this level? Sound, light, alternating current, planetary orbits, and seasonal temperature all rise and fall periodically. The instants when a periodic quantity equals some threshold are precisely the roots of a trigonometric equation.
Engineering. An AC voltage $V = V_0 \sin(\omega t)$ crosses zero, peaks, and returns on a fixed schedule; solving for $t$ tells a circuit when to switch.
Navigation and astronomy. Predicting when a satellite or the Sun reaches a given angle above the horizon is a trigonometric equation in disguise.
The general-solution payoff. A single high tide is useless to a port. The general solution is what turns one answer into a timetable.
This is why the periodic copies are not optional decoration. Drop them and you have solved the equation for today only.
Tripping Points to Avoid
Mistake 1: Reporting only the principal solution
Where it slips in: When the question asks for the general solution but a familiar angle pops out first.
Don't do this: Writing $x = \dfrac{\pi}{6}$ and stopping because $\sin\dfrac{\pi}{6} = \dfrac{1}{2}$.
The correct way: Apply the general-solution formula every time the word "general" appears: $x = n\pi + (-1)^n \dfrac{\pi}{6}$.
The first-instinct error students reach for is treating a trigonometric equation like an algebraic one with a unique root — the periodic nature of the function is exactly the habit the general-solution formula is built to fix.
Mistake 2: Using the wrong formula for the function
Where it slips in: Reaching for the sine formula's $(-1)^n$ when the equation is in cosine.
Don't do this: Writing $x = n\pi + (-1)^n\alpha$ for $\cos x = \cos\alpha$.
The correct way: Match the function to its own formula — cosine uses $x = 2n\pi \pm \alpha$, tangent uses $x = n\pi + \alpha$.
The memorizer who learns one formula and assumes it covers all three loses marks the moment the equation switches function. The second-guesser, by contrast, often solves correctly then erases the periodic term out of doubt — both fixes come from trusting the formula that matches the function.
Mistake 3: Dividing both sides by a trig term
Where it slips in: Equations like $\sin x \cos x = \sin x$, where dividing by $\sin x$ looks harmless.
Don't do this: Cancelling $\sin x$ to get $\cos x = 1$, which silently discards all solutions where $\sin x = 0$.
The correct way: Move everything to one side and factor: $\sin x(\cos x - 1) = 0$, then solve each factor.
Key Takeaways
A trigonometric equation has infinitely many solutions, captured by a general-solution formula with integer $n$.
$\sin x = \sin\alpha \Rightarrow x = n\pi + (-1)^n\alpha$; $\cos x = \cos\alpha \Rightarrow x = 2n\pi \pm \alpha$; $\tan x = \tan\alpha \Rightarrow x = n\pi + \alpha$.
The principal solution is the version restricted to $[0, 2\pi)$.
Never divide by a trig term — factor instead, or you lose solutions.
Match the formula to the function; the three are not interchangeable.
To take trigonometric equations further with a teacher, explore Bhanzu's trigonometry tutor sessions, a high school math tutor for exam-level practice, or live math classes online with peers from 20+ countries.
A Practical Next Step
Practice these to solidify your understanding: solve $\cos x = \dfrac{1}{\sqrt{2}}$ for its general solution, then $2\sin^2 x - 1 = 0$. If you get stuck choosing the formula, come back to the general-solution table above. Want a live Bhanzu trainer to walk through more problems? Book a free demo class.
Read More
Trigonometric identities — the identities that simplify equations before you solve them.
Sum and difference identities — for equations mixing compound angles.
Principal value of trigonometric functions — how the principal solution is chosen.
Domain and range of trigonometric functions — why some equations have no solution.
Trigonometric table — the special-angle values you reach for in step 2.
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