Sine rule calculator with working

• Missing side: $b = 10$ cm, $B = 60°$, $A = 40°$. $a = \dfrac{10 \times \sin 40°}{\sin 60°} \approx 7.42$ cm. The third angle is $180 - 40 - 60 = 80°$.
• Missing angle: $a = 8$ cm, $b = 10$ cm, $B = 70°$. $\sin A = \dfrac{8 \times \sin 70°}{10} \approx 0.752$, so $A \approx 48.7°$.

Label the angles with capital letters and the sides opposite them with the same lowercase letter: side $a$ is opposite angle $A$. Getting the pairs right is the key step; mixing them up gives a wrong answer that still looks reasonable.

When you find an angle with the sine rule, there can be two answers, because $\sin\theta = \sin(180° - \theta)$. If the obtuse angle $180° - A$ still fits in the triangle (it plus the known angle is less than 180°), both triangles are possible. The calculator shows the second answer when it exists; use the diagram (is the angle clearly acute or obtuse?) to choose.

• Sine rule: you know a side and its opposite angle (a matching pair).
• Cosine rule: you know two sides and the angle between them, or all three sides.

Our cosine rule calculator handles the second case. For a triangle's area from two sides and the included angle, use $\tfrac{1}{2}ab\sin C$.

• Pairing a side with the wrong angle.
• Calculator in radians.
• Missing the second (obtuse) answer when a question asks for all possible values.

Practise with our sine rule questions.

• Write the rule with the values in before rearranging.
• When finding an angle, flip the rule to $\tfrac{\sin A}{a} = \tfrac{\sin B}{b}$ so the unknown is on top.
• The sine rule is Higher tier at GCSE and Extended tier at IGCSE.

Sine rule questions often come with bearings. A ship sails between two ports, and the question gives the angles at each port and one distance; drawing the triangle and marking the angles from north turns it into a standard sine rule problem. Surveyors use the same idea (triangulation) to find the distance to a point they can't reach, by measuring a baseline and the two angles to the point.