How Do You Use the Sine Rule for a Non-Right Triangle?

• French B
• Geography
• German A
• German A Lit
• German ab initio
• German B
• Global Politics
• History
• History (First Exam 2028)
• Mathematics Analysis and Approaches (AA)
• Mathematics Applications & Interpretation (AI)

• Philosophy
• Physics
• Psychology
• Psychology (First Exam 2027)
• Spanish A
• Spanish A Lit
• Spanish ab initio
• Spanish B
• Sports, exercise and health science (SEHS)
• Theory of Knowledge (TOK)
• World Religions

• MYP English Lang & Lit
• MYP Extended Mathematics
• MYP History

• MYP Individuals & Societies
• MYP Integrated Sciences
• MYP Physics

• MYP Standard Mathematics

The sine rule works because the ratio of each side length to the sine of its opposite angle is constant within the triangle. It is especially useful for non-right triangles when at least one complete opposite side-angle pair is known.

Do not use the sine rule when no complete opposite pair is known. If three sides, or two sides and the included angle, are given, use the cosine rule instead. If two angles are known, first find the third using $A+B+C=180^\circ$.

Worked Example

Suppose $A=42^\circ$, $B=71^\circ$, and $a=8$. Find $b$.

First identify the opposite pairs: $a=8$ is opposite $A=42^\circ$, while $b$ is opposite $B=71^\circ$. Then write only the required two fractions:

Rearrange for $b$:

Therefore, $b=11.3$ units to three significant figures.

A common misconception is pairing a side with an adjacent angle. Each side must be paired with the angle directly opposite it. When finding an angle from two sides and one angle, also check for the ambiguous case: a second triangle may exist because $\sin \theta=\sin(180^\circ-\theta)$. After finding $\theta$, test $180^\circ-\theta$ against the angle sum and given measurements; do not automatically report both values.

For IB Mathematics AA SL, this is SL 3.2: right-angled and non-right-angled trigonometry and may appear on Paper 1 or Paper 2; the ambiguous case is specifically SL 3.5. For a “find” or “calculate” question, show the substituted sine-rule equation before giving the final answer; a calculator answer alone may not earn full method marks.