A basic trigonometric equation asks for all angles that make a trigonometric statement true.
For example:
\[
2\cos x=\sqrt2
\]
First isolate the trigonometric function:
\[
\cos x=\frac{\sqrt2}{2}
\]
Then use the unit circle to find all matching angles in the required interval.
1. Sine equations
If:
\[
\sin u=k
\]
then the general solutions are:
\[
u=\alpha+2\pi n
\]
\[
u=\pi-\alpha+2\pi n
\]
where \(\alpha=\arcsin(k)\) and \(n\in\mathbb{Z}\).
2. Cosine equations
If:
\[
\cos u=k
\]
then the general solutions are:
\[
u=\pm\alpha+2\pi n
\]
where \(\alpha=\arccos(k)\) and \(n\in\mathbb{Z}\).
3. Tangent equations
If:
\[
\tan u=k
\]
then the general solution is:
\[
u=\alpha+\pi n
\]
where \(\alpha=\arctan(k)\) and \(n\in\mathbb{Z}\).
Tangent repeats every \(\pi\), not every \(2\pi\).
4. Transformed trig equations
Many trig equations can be written as:
\[
A f(Bx-C)+D=K
\]
First isolate the trig function:
\[
f(Bx-C)=\frac{K-D}{A}
\]
Then solve for the argument \(u=Bx-C\), and finally solve for \(x\).
5. Example
Solve:
\[
2\cos x=\sqrt2,\qquad 0\le x\le 2\pi
\]
Divide by 2:
\[
\cos x=\frac{\sqrt2}{2}
\]
On the unit circle, cosine is \(\frac{\sqrt2}{2}\) in Quadrant I and Quadrant IV:
\[
x=\frac{\pi}{4},\quad x=\frac{7\pi}{4}
\]
6. Formula summary
| Equation |
Base angle |
General solution |
| \(\sin u=k\) |
\(\alpha=\arcsin(k)\) |
\(u=\alpha+2\pi n\) or \(u=\pi-\alpha+2\pi n\) |
| \(\cos u=k\) |
\(\alpha=\arccos(k)\) |
\(u=\pm\alpha+2\pi n\) |
| \(\tan u=k\) |
\(\alpha=\arctan(k)\) |
\(u=\alpha+\pi n\) |
7. Domain checks
- \(\sin u=k\) has real solutions only when \(-1\le k\le 1\).
- \(\cos u=k\) has real solutions only when \(-1\le k\le 1\).
- \(\tan u=k\) has real solutions for every real \(k\).
8. Common mistakes
- Do not forget the second sine or cosine solution on the unit circle.
- Use period \(2\pi\) for sine and cosine, but period \(\pi\) for tangent.
- When the argument is \(2x\), solve for \(2x\) first, then divide by 2.
- Always check whether the final \(x\)-values lie inside the required interval.
- Remember that open intervals exclude endpoints.
