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Algebra readiness

Use radians and triangle ratios · Diagrams

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General triangle side and angle names

Match a lowercase side with the opposite uppercase angle. These rules apply to general triangles; no right angle is assumed.

ABCabc
Each side is opposite its matching angle. The sketch supplies no side lengths.

Match opposite sides and angles

Opposite angle A

a=BCa=BC

Opposite angle B

b=CAb=CA

Opposite angle C

c=ABc=AB

Cosine rule: use the included angle

Use two sides and their included angle, or use all three sides to find an angle. Choose the form with the required side opposite its matching angle.

For side a, opposite A

a2=b2+c2−2bccos⁡A.\begin{aligned}a^2&=b^2+c^2\\&\quad-2bc\cos A.\end{aligned}

For side b, opposite B

b2=a2+c2−2accos⁡B.\begin{aligned}b^2&=a^2+c^2\\&\quad-2ac\cos B.\end{aligned}

For side c, opposite C

c2=a2+b2−2abcos⁡C.\begin{aligned}c^2&=a^2+b^2\\&\quad-2ab\cos C.\end{aligned}

Try the included-angle example

Sine rule: keep each opposite pair together

sin⁡Aa=sin⁡Bb=sin⁡Cc.\begin{aligned}\frac{\sin A}{a}&=\frac{\sin B}{b}\\&=\frac{\sin C}{c}.\end{aligned}

If inverse sine gives an angle, check whether its supplementary angle also satisfies the triangle's other data.

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General triangle ABC has angle labels A,B,C and opposite sides a=BC,b=CA,c=AB. No right angle is assumed.
General triangle ABC has angle labels A,B,C and opposite sides a=BC,b=CA,c=AB. No right angle is assumed.
General triangle ABC has angle labels A,B,C and opposite sides a=BC,b=CA,c=AB. No right angle is assumed.
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