HG22 - Sine and cosine rules

Sine rule

A triangle has angles $40^\circ$ and $65^\circ$, with side $8\,\mathrm{cm}$ opposite $40^\circ$. How can you find the side opposite $65^\circ$ without a right angle?

Label vertices $A,B,C$ and their opposite sides $a,b,c$: $a=BC$, $b=CA$, $c=AB$. The letters are a pairing convention, not a requirement that $c$ is a hypotenuse. Only a right-angled triangle has a hypotenuse.

[DIAGRAM: asset_name: Opposite pairs and obtuse angles; asset_slug: general-triangle-pairs; caption: Match each side to the angle opposite it; reflected unit-circle points explain supplementary-angle signs.; description: An exact scalene triangle labelled A B C with opposite sides a b c, beside a unit semicircle showing 60 and 120 degree radii with equal positive heights and opposite horizontal coordinates.]

For an acute triangle, dropping a perpendicular from $C$ to $AB$ gives the same height as $b\sin A$ and $a\sin B$. Thus $b\sin A=a\sin B$, which rearranges to $a/\sin A=b/\sin B$. Repeating with another height brings in the third pair.

For angles between $90^\circ$ and $180^\circ$, use the unit-circle interpretation: sine is the vertical coordinate, cosine the horizontal coordinate. Reflecting a radius across the vertical axis gives the supplementary angle $180^\circ-\theta$. Its height is unchanged and its horizontal coordinate changes sign:

So a triangle's obtuse angle has positive sine and negative cosine. An outside altitude and the equal supplementary sines extend the sine rule to obtuse triangles too.

A complete opposite pair provides the common ratio; use it to calculate the missing member of a second pair.

Change to finding an angle. Inverse sine returns the acute candidate; an obtuse supplement may need checking, as the third part explains.

The side ordering and angle sum give geometric checks on an inverse-sine result.

Use a fresh complete pair.

Cosine rule

If no complete opposite pair is known, the sine rule may not start the problem. SAS means two sides and their included angle; SSS means all three sides. The cosine rule handles both.

The included angle lies between the two known sides. In $a^2=b^2+c^2-2bc\cos A$, $A$ is between $b$ and $c$, and $a$ lies opposite $A$. You may relabel the formula to fit any vertex.

The correction term explains the link to Pythagoras. At $90^\circ$, cosine is zero. For an acute included angle, subtract a positive quantity; for an obtuse angle, its cosine is negative, so the correction increases the opposite side's square.