Basic trigonometric equations: complete solution families
Solve sine, cosine, tangent, and cotangent equations using complete families, principal-value conventions, and interval restrictions.
Content updated: October 22, 2026
A trigonometric value usually corresponds to many angles. Solving an equation therefore requires more than one calculator value: find every solution family and retain the domain restrictions.
Learning goals: solve the four basic trigonometric equations, distinguish principal values from general solutions, and select solutions in an interval.
Prerequisites: the unit circle, radians, periods, and special trigonometric values.
On this page: The idea · Method · Examples · Practice · Further learning
1. Domains and angle units
Throughout this lesson, $x$ is a real angle measured in radians and $k\in\mathbb Z$. If using degrees, replace $2\pi$ by $360^\circ$ and $\pi$ by $180^\circ$ consistently throughout; do not mix the two units.
Since $-1\le\sin x,\cos x\le1$, the equations $\sin x=a$ and $\cos x=a$ have no real solutions when $|a|>1$. The function $\tan x$ is defined only when $\cos x\ne0$, while $\cot x$ requires $\sin x\ne0$. On these domains, tangent and cotangent attain every real value.
2. Four general-solution formulas
The two sine branches correspond to two points sharing a y-coordinate on the circle; the cosine branches share an x-coordinate. For values $1$ or $-1$, the branches may coincide as one family. For the value $0$, the solutions can be combined into a family with period $\pi$.
The conditions on the right ensure that the expression involving $\alpha$ is defined. For example, $\tan x=\tan\alpha$ implies $\sin(x-\alpha)=0$ by multiplying by two cosines different from $0$. The converse follows from period $\pi$ and preserves the domain.
For principal angles, use $\arcsin a\in[-\pi/2;\pi/2]$, $\arccos a\in[0;\pi]$, and $\arctan a\in(-\pi/2;\pi/2)$. Here $\operatorname{arccot}a$ is defined as the unique angle in $(0;\pi)$ having cotangent $a$. A principal angle is only an intermediate step toward the complete solution families.
3. Eight basic examples
Example 1. Sine of a special angle
The equation $\sin x=1/2=\sin(\pi/6)$ gives $x=\pi/6+2k\pi$ or $x=5\pi/6+2k\pi$. Writing only $x=\pi/6$ omits the second branch and all full turns.
Example 2. A nonspecial sine value
Solve $\sin x=1/3$. Set $\alpha=\arcsin(1/3)$. Then $\sin\alpha=1/3$, and the solutions are $x=\alpha+2k\pi$ or $x=\pi-\alpha+2k\pi$. No rounding of $\alpha$ is needed for an exact answer.
Example 3. Cosine of a special angle
The equation $\cos x=1/2=\cos(\pi/3)$ gives $x=\pi/3+2k\pi$ or $x=-\pi/3+2k\pi$. These families may be written compactly as $x=\pm\pi/3+2k\pi$.
Example 4. A nonspecial cosine value
Solve $\cos x=1/3$. Set $\beta=\arccos(1/3)$. The equation becomes $\cos x=\cos\beta$, so $x=\pm\beta+2k\pi$. The cosine function must be retained in this transformation.
Example 5. Tangent of a given angle
Assuming $\cos\alpha\ne0$, the equation $\tan x=\tan\alpha$ has solutions $x=\alpha+k\pi$. For example, $\alpha=\pi/4$ gives $x=\pi/4+k\pi$. The angle $\alpha$ must not be omitted from the formula.
Example 6. Tangent equal to 3
Solve $\tan x=3$. Setting $\gamma=\arctan3$ gives $x=\gamma+k\pi$. Since $\gamma\in(-\pi/2;\pi/2)$, every solution has cosine different from $0$ and belongs to the domain.
Example 7. Cotangent of a special angle
The equation $\cot x=1/\sqrt3=\cot(\pi/3)$ gives $x=\pi/3+k\pi$. Every solution has sine different from $0$.
Example 8. Cotangent equal to 3
Solve $\cot x=3$. Set $\delta=\operatorname{arccot}3$, or use $\delta=\arctan(1/3)$ since this value lies in $(0;\pi/2)$. The solutions are $x=\delta+k\pi$.
4. Practice exercises
- Solve $\sin x=0$ and $\cos x=-1$.
- Solve $\sin(2x)=1/2$ on $[0;\pi)$.
- Solve $\tan x=-1$, and decide whether $\sin x=3/2$ has a solution.
Show hints and answers
- $\sin x=0$ gives $x=k\pi$; $\cos x=-1$ gives $x=\pi+2k\pi$.
- From $2x=\pi/6+2k\pi$ or $2x=5\pi/6+2k\pi$, we obtain $x=\pi/12+k\pi$ or $x=5\pi/12+k\pi$. In the given interval, only $\pi/12,5\pi/12$ occur.
- $x=-\pi/4+k\pi$. The sine equation has no solution because $3/2>1$.
5. Common mistakes
Do not omit the reference angle in tangent or cotangent formulas. Do not use period $2\pi$ in a way that captures only some tangent or cotangent solutions. When dividing an angle expression such as $2x$, divide the period as well. Check the domain first and filter the interval after finding the general solution.
6. Related lessons
- The unit circle (English lesson)
- Representing angles on the unit circle (English lesson)
- Trigonometric identities (English lesson)
- Equations of the form a sin x + b cos x = c (English lesson)
- Quadratic equations in a trigonometric function (English lesson)
7. Resources from the original article
Original source resources
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