A trigonometric equation is an equation involving trigonometric functions such as
\(\sin x\), \(\cos x\), or \(\tan x\). Because these functions repeat, a trig equation
often has many solutions. A solver must find the general pattern and then keep only the
solutions inside the selected interval.
1. Periods of the main functions
The basic periods are:
\[
\begin{aligned}
\sin(x+2\pi)&=\sin x,\\
\cos(x+2\pi)&=\cos x,\\
\tan(x+\pi)&=\tan x.
\end{aligned}
\]
In degrees, these periods are:
\[
\begin{aligned}
\sin(x+360^\circ)&=\sin x,\\
\cos(x+360^\circ)&=\cos x,\\
\tan(x+180^\circ)&=\tan x.
\end{aligned}
\]
2. Solving \(\sin x=c\)
If \(-1\le c\le1\), let:
\[
\begin{aligned}
\alpha=\arcsin(c).
\end{aligned}
\]
Then the general solutions are:
\[
\begin{aligned}
x&=\alpha+2\pi k,\\
x&=\pi-\alpha+2\pi k,
\end{aligned}
\qquad k\in\mathbb{Z}.
\]
If \(|c|>1\), there is no real solution because sine values must stay between \(-1\) and \(1\).
3. Solving \(\cos x=c\)
If \(-1\le c\le1\), let:
\[
\begin{aligned}
\alpha=\arccos(c).
\end{aligned}
\]
The general solutions are:
\[
\begin{aligned}
x=\pm\alpha+2\pi k,
\qquad k\in\mathbb{Z}.
\end{aligned}
\]
4. Solving \(\tan x=c\)
Tangent has period \(\pi\), so:
\[
\begin{aligned}
x=\arctan(c)+\pi k,
\qquad k\in\mathbb{Z}.
\end{aligned}
\]
Unlike sine and cosine, tangent can take any real value.
5. Worked example: quadratic in sine
Solve:
\[
\begin{aligned}
2\sin^2x-\sin x-1=0
\end{aligned}
\]
on the interval \([0,2\pi)\).
Let:
\[
\begin{aligned}
u=\sin x.
\end{aligned}
\]
Then:
\[
\begin{aligned}
2u^2-u-1=0.
\end{aligned}
\]
Factor:
\[
\begin{aligned}
2u^2-u-1&=(2u+1)(u-1).
\end{aligned}
\]
So:
\[
\begin{aligned}
u&=1
\qquad\text{or}\qquad
u=-\frac12.
\end{aligned}
\]
Replace \(u\) with \(\sin x\):
\[
\begin{aligned}
\sin x&=1
\qquad\text{or}\qquad
\sin x=-\frac12.
\end{aligned}
\]
In \([0,2\pi)\):
\[
\begin{aligned}
\sin x=1&\Rightarrow x=\frac{\pi}{2},\\
\sin x=-\frac12&\Rightarrow x=\frac{7\pi}{6},\frac{11\pi}{6}.
\end{aligned}
\]
Therefore:
\[
\begin{aligned}
\boxed{x=\frac{\pi}{2},\frac{7\pi}{6},\frac{11\pi}{6}}.
\end{aligned}
\]
6. Unit-circle interpretation
The unit circle represents an angle \(x\) using the point:
\[
\begin{aligned}
(\cos x,\sin x).
\end{aligned}
\]
Therefore:
\[
\begin{aligned}
\sin x&=\text{the y-coordinate},\\
\cos x&=\text{the x-coordinate}.
\end{aligned}
\]
For example, \(\sin x=-1/2\) means the unit-circle point has y-coordinate \(-1/2\),
which occurs in Quadrants III and IV.
7. Interval solutions
General formulas describe infinitely many angles, but most calculators ask for solutions in a chosen interval.
For example:
\[
\begin{aligned}
\sin x=\frac12
\end{aligned}
\]
has the general solutions:
\[
\begin{aligned}
x&=\frac{\pi}{6}+2\pi k,\\
x&=\frac{5\pi}{6}+2\pi k.
\end{aligned}
\]
But in the interval \([0,2\pi)\), only:
\[
\begin{aligned}
x=\frac{\pi}{6},\frac{5\pi}{6}
\end{aligned}
\]
are reported.
8. Equations requiring numerical methods
Some trig equations are not easy to solve by a single inverse-trig rule. Examples include:
\[
\begin{aligned}
\sin(2x)&=\cos x,\\
\sin x+\cos x&=1,\\
\tan x&=\sin x+1.
\end{aligned}
\]
For these, the calculator can rewrite the problem as:
\[
\begin{aligned}
F(x)=\text{LHS}(x)-\text{RHS}(x)=0.
\end{aligned}
\]
Then it searches for zeros of \(F(x)\) in the selected interval and verifies each candidate.
9. Graph interpretation
A trigonometric equation:
\[
\begin{aligned}
\text{LHS}(x)=\text{RHS}(x)
\end{aligned}
\]
can be graphed as:
\[
\begin{aligned}
y_1&=\text{LHS}(x),\\
y_2&=\text{RHS}(x).
\end{aligned}
\]
Solutions occur where the two curves intersect. If the two curves overlap for an entire interval,
every point in that interval is a solution.
10. Domain issues
Sine and cosine are defined for every real \(x\). Tangent and reciprocal functions have restrictions:
\[
\begin{aligned}
\tan x&=\frac{\sin x}{\cos x},\quad \cos x\ne0,\\
\cot x&=\frac{\cos x}{\sin x},\quad \sin x\ne0,\\
\sec x&=\frac{1}{\cos x},\quad \cos x\ne0,\\
\csc x&=\frac{1}{\sin x},\quad \sin x\ne0.
\end{aligned}
\]
A numerical solver must avoid false roots near undefined points.
11. Formula summary
The table uses plain-text formulas in the cells to avoid raw LaTeX issues in narrow layouts.
12. Common mistakes
- Reporting only one solution when sine or cosine has two solutions in one cycle.
- Forgetting the period \(2\pi k\) or \(\pi k\) in the general solution.
- Mixing radians and degrees.
- Including \(2\pi\) in \([0,2\pi)\), even though the right endpoint is excluded.
- Accepting false roots near tangent asymptotes.
- Rounding too early before checking the original equation.
Key idea: solve the trig pattern, use periodicity to generate all angles, keep only the values in the interval, and verify each one.
