The Law of Sines relates the sides of a triangle to the sines of their opposite angles.
In standard notation, side \(a\) is opposite angle \(A\), side \(b\) is opposite angle \(B\),
and side \(c\) is opposite angle \(C\).
\[
\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}
\]
1. When to use the Law of Sines
The Law of Sines is especially useful when you know:
- two angles and one side, called AAS or ASA;
- two sides and a non-included opposite angle, called SSA.
If you know all three sides or two sides with the included angle, you normally need the Law of Cosines first.
2. Solving with two angles and one side
First find the missing angle:
\[
A+B+C=180^\circ
\]
Then use the Law of Sines to find the missing sides. For example:
\[
\frac{a}{\sin A}=\frac{b}{\sin B}
\]
\[
b=\frac{a\sin B}{\sin A}
\]
3. The SSA ambiguous case
SSA can sometimes produce two different triangles, one triangle, or no triangle.
Suppose \(A\), \(a\), and \(b\) are known. Then:
\[
\frac{a}{\sin A}=\frac{b}{\sin B}
\]
\[
\sin B=\frac{b\sin A}{a}
\]
If the computed sine value is greater than \(1\), no triangle is possible. If it is between \(0\)
and \(1\), two angle choices may be possible:
\[
B=\arcsin(\text{value})
\]
\[
B=180^\circ-\arcsin(\text{value})
\]
Each choice must still leave a positive third angle.
4. Example
Suppose \(A=45^\circ\), \(B=65^\circ\), and \(a=12\). Then:
\[
C=180^\circ-45^\circ-65^\circ=70^\circ
\]
\[
\frac{12}{\sin45^\circ}=\frac{b}{\sin65^\circ}
\]
\[
b=\frac{12\sin65^\circ}{\sin45^\circ}
\]
5. Formula summary
| Goal |
Formula |
Meaning |
| Law of Sines |
\(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\) |
Compares each side with the sine of its opposite angle |
| Find side \(b\) |
\(b=\dfrac{a\sin B}{\sin A}\) |
Use when \(a\), \(A\), and \(B\) are known |
| Find angle \(B\) |
\(\sin B=\dfrac{b\sin A}{a}\) |
Use with care in SSA cases |
| Angle sum |
\(A+B+C=180^\circ\) |
Every triangle has angle sum \(180^\circ\) |
6. Common mistakes
- Match each side with its opposite angle: \(a\) with \(A\), \(b\) with \(B\), and \(c\) with \(C\).
- Do not use the Law of Sines for SSS before finding an angle with the Law of Cosines.
- In SSA, always check whether a second triangle is possible.
- If the angle sum reaches \(180^\circ\) or more, the triangle is impossible.
- Keep angle units consistent: degrees with degrees, radians with radians.
