Triangle Ratios and Rules
Triangle Ratios and Rules
Choose the angle first, then identify opposite, adjacent and hypotenuse. The hypotenuse is opposite the right angle.
\(\sin\theta=\dfrac{\text{opposite}}{\text{hypotenuse}},\quad\cos\theta=\dfrac{\text{adjacent}}{\text{hypotenuse}},\quad\tan\theta=\dfrac{\text{opposite}}{\text{adjacent}}\)
Side \(a\) is opposite \(A\), side \(b\) opposite \(B\), and side \(c\) opposite \(C\). Thus \(C=\angle ACB\).
| Given information | Useful rule |
|---|---|
| A side and its opposite angle, with another side or angle | \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\) |
| Two sides and their included angle | \(c^2=a^2+b^2-2ab\cos C\) |
| Three sides | \(\cos C=\dfrac{a^2+b^2-c^2}{2ab}\) |
| Two sides and their included angle, for area | \(\text{Area}=\dfrac12ab\sin C\) |
Check the angle unit and whether the angle lies between the sides used in the area formula. Keep full calculator precision until the final answer.
The sine and cosine rules also hold in a right-angled triangle. The area formula works for an obtuse included angle because \(\sin(180^\circ-C)=\sin C\). Choose the included angle between the two sides used.
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