Quick questions on Right-angled triangle trigonometry for HSC Maths Advanced: the sine, cosine and tangent ratios, finding sides and angles, the exact ratios of 30, 45 and 60 degrees, angles of elevation and depression, and compass and true bearings

Because any two right-angled triangles with the same acute angle $\theta$ are similar (the AA test), the ratio of any two sides depends only on $\theta$, not on the size of the triangle. Those fixed ratios are the trigonometric functions:

When the unknown is a side, choose the ratio that pairs the unknown with a known side, write it with the unknown on top, then make the unknown the subject in one step. If the unknown is on top, the last step is a multiplication; if the unknown sits in the denominator (the known side is the opposite or adjacent and you want the hypotenuse), it is a division. Keeping the unknown on top from the start avoids the cross-multiplication slip that loses easy marks.

When two sides are known and the unknown is the angle, work out from those two sides which ratio is fixed, write it, then apply the matching inverse function. For example, if the opposite and adjacent are known, $\tan\theta$ is fixed, so $\theta = \tan^{-1}\!\left(\dfrac{O}{A}\right)$. The inverse-trig keys ($\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$) return the acute angle directly, which is all a right-angled triangle ever needs.