SL 3.2—Triangle trigonometry

Pythagoras links the sides of a right triangle: a²+b²=c², where c is opposite the right angle. Sine, cosine and tangent link an acute angle to the opposite, adjacent and hypotenuse sides.

Label the right angle and the chosen angle first. Use Pythagoras when all you need is a side length; use a trigonometric ratio when an angle is involved. The inverse ratio finds the angle after the side relationship is formed.

If the opposite side is 6 and the hypotenuse 10, sin θ=0.6, so θ≈36.9°. The same triangle gives the adjacent side √64=8, providing a consistency check.

SOHCAHTOA depends on the chosen angle. Do not use the hypotenuse as the adjacent side or round before the final step.

For any triangle, $a/\sin A=b/\sin B=c/\sin C$, $c^2=a^2+b^2-2ab\cos C$, and $Area=\tfrac12ab\sin C$. Example: with $a=7$, $b=9$, $C=60^\circ$, $c^2=67$ and the area is $63\sqrt3/4$ square units. Choose the sine rule for a known opposite side–angle pair, the cosine rule for SAS or SSS, and note that the ambiguous sine-rule case is not included at SL.