Sine and cosine of obtuse angles

An obtuse angle lies between \(90^\circ\) and \(180^\circ\). For acute \(\theta\), the supplementary angle \(180^\circ-\theta\) has the same positive sine and the opposite cosine: \(\sin(180^\circ-\theta)=\sin\theta\), \(\cos(180^\circ-\theta)=-\cos\theta\).

The reflected radii \(OP\) and \(OQ\) have the same upward height but opposite horizontal components. Taking each component as a fraction of the radius explains the signs. The diagram illustrates angles measured anticlockwise from \(OX\).

Example

\(\sin120^\circ=\sin60^\circ=\dfrac{\sqrt3}{2}\), while \(\cos120^\circ=-\cos60^\circ=-\dfrac12\). If a triangle angle has sine \(0.6\), the candidates are approximately \(36.9^\circ\) and \(143.1^\circ\); the other triangle information determines which is possible.

Use this supplementary-angle relationship in triangle calculations. A calculator’s inverse sine gives the acute candidate for a positive value; it does not automatically return a possible obtuse triangle angle.