Trigonometry in the Right Triangle

Decide by the sides involved. Sine links opposite and hypotenuse (sin α = opp/hyp), cosine links adjacent and hypotenuse (cos α = adj/hyp), tangent links the two legs (tan α = opp/adj). So look at which two sides occur in your task (given plus required), and pick the function that contains exactly these two. Example: ladder length (hypotenuse) and wall height (opposite side) given, angle required: sine. Ground distance and wall height: tangent. The classic mnemonic is SOH-CAH-TOA. A quick sketch with the angle marked helps prevent swapping the opposite and adjacent legs.

The hypotenuse is fixed: the longest side, always opposite the right angle. Opposite and adjacent side, in contrast, are named relative to the angle considered and swap roles when you take the other acute angle. The opposite side lies across from the angle (it does not touch it), the adjacent side lies next to the angle but is not the hypotenuse. Practical approach: put a finger on the angle; the side the finger does not touch is the opposite side; of the two touched sides, the leg (not the hypotenuse) is the adjacent side. Exactly this assignment is the most frequent source of errors, so a labelled sketch before every calculation pays off.

First form the appropriate side ratio and then apply the inverse function: arcsin, arccos or arctan (on the calculator $\sin{}^{-1}$, $\cos{}^{-1}$, tan⁻¹). Example: opposite 3, adjacent 4: $\tan{} \alpha{} =\frac{ 3}{4} = 0.75$, so $\alpha{} = \arctan{}(0.75) \approx{} 36.9$°. The second acute angle follows immediately from the angle sum: $\beta{} = 90$° $- \alpha{} \approx{} 53.1$°. Check the angle mode first (DEG for degree values). Important: the inverse function needs the ratio of two sides, not a single length. And as a check: the sine of the result must reproduce the original ratio, here sin 36.9° ≈ 0.6 = 3/5 ✓.

Almost always the wrong angle mode is behind it. Calculators know degrees (DEG) and radians (RAD). If you enter sin 30 in RAD mode, the calculator interprets 30 as 30 radians and returns about −0.988 instead of the expected 0.5. Therefore check the mode indicator on the display before trigonometric calculations. Rule of thumb for school: geometry tasks with degree values in DEG; calculus with trigonometric functions (derivatives, integrals) requires RAD, because (sin x)′ = cos x holds only in radians. A quick self-test after switching: sin 30° must give exactly 0.5, sin 90° exactly 1.

You should know three relations. First: $\tan{} \alpha{} =\frac{ \sin{} \alpha{}}{\cos{}} \alpha{}$, since (opp/hyp)/(adj/hyp) cancels to opp/adj. Second the trigonometric Pythagoras: $\sin{}^{2}\alpha{} + \cos{}^{2}\alpha{} = 1$, because opp² + adj² = hyp²; with it you compute one value from the other, e.g. $\cos{} \alpha{} = \sqrt{(1 - \sin{}^{2}\alpha{})}$ for acute angles. Third the complement relation: cos α = sin(90° − α), the cosine is the sine of the complementary angle, hence its name (complementi sinus). Example: sin 30° = 0.5 and cos 30° ≈ 0.866; check: $0.25 + 0.75 = 1$ ✓ and tan 30° = 0.5/0.866 ≈ 0.577 ✓.