How Do You Use the Sine Rule and Cosine 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

Suppose $A=42^\circ$, $a=8$, and $B=65^\circ$. To find $b$, pair each side with its opposite angle:

Using the Cosine Rule

To find a side, use

If $a=7$, $b=9$, and the included angle $C=60^\circ$, then

To find an angle from three sides, rearrange:

A Second Cosine-Rule Case

If the three sides are $a=5$, $b=7$, and $c=8$, find $C$ by substituting into the angle form:

Therefore,

This example shows why matching the side and angle labels matters: $c$ must be opposite $C$. Before calculating, identify whether the known angle is included between the two known sides. If it is, use the cosine rule. If you have a complete opposite pair, use the sine rule. You can then use $A+B+C=180^\circ$ to find a remaining angle.

A common misconception is that the sine rule works whenever two sides are known. It requires a known opposite side-angle pair; SAS normally requires the cosine rule.

IB Exam Technique

For IB Math AI SL, keep the GDC in degree mode, write the correct formula and substitution before the decimal answer, and retain full precision until the final rounding. The ambiguous sine-rule case is not assessed at SL.