Ratios and opposite pairs

Use degree mode for angles in degrees. In a right-angled triangle, identify opposite, adjacent and hypotenuse relative to the chosen acute angle.

For acute \(\theta\), \(\sin(180^\circ-\theta)=\sin\theta\), \(\cos(180^\circ-\theta)=-\cos\theta\), and \(\tan(180^\circ-\theta)=-\tan\theta\). An obtuse triangle angle has positive sine but negative cosine and tangent.

In any triangle, pair each side with its opposite angle: \(\dfrac a{\sin A}=\dfrac b{\sin B}=\dfrac c{\sin C}\). Use the sine rule when a complete opposite side-angle pair is known. The angles of a triangle sum to \(180^\circ\).