Right-Triangle Cosine Ratio (CAH)

CAH: Cosine is Adjacent over Hypotenuse. Like all the trigonometric ratios, it works because the shape of a right triangle is fixed by one acute angle alone: every right triangle with a 60° angle has an adjacent leg exactly half its hypotenuse, whether it is drawn on a napkin or laid out across a field. Similar triangles keep the ratio constant, and that constancy is what turns an angle into a length. The name is a contraction of complementi sinus, the sine of the complement, because $\cos\theta = \sin(90° - \theta)$: cosine is not a second idea but the same idea viewed from the other acute corner.

A worked instance you can check against a wall. A ladder is meant to stand at about 75° to the ground, the familiar one-out-for-four-up rule. A 6 m ladder therefore has its feet $6\cos 75° = 1.55$ m from the base of the wall, and reaches 5.8 m up it. Run the other way, $\theta = \arccos(a/h)$: a 6 m ladder set 2.0 m out is standing at $\arccos(0.333) = 70.5°$, flatter than it should be.

Three values are worth knowing cold: $\cos 0° = 1$, $\cos 60° = 0.5$, $\cos 90° = 0$. Beyond the triangle, cosine is the universal "how much of this points that way" operator. The component of a force along a direction is $F\cos\theta$; the useful part of an alternating current is the power factor $\cos\varphi$; the projection of any vector onto any axis is a cosine. On the unit circle it is simply the x-coordinate, and the identity $\sin^2\theta + \cos^2\theta = 1$ is the Pythagorean theorem on a triangle of hypotenuse 1.

Where it goes wrong. The most frequent error is picking the wrong leg: the adjacent side is the one touching the angle that is not the hypotenuse, and since the hypotenuse also touches the angle, that phrasing is exactly where people slip. The second is degree-versus-radian mode. This solver handles the conversion, but a phone calculator left in radians returns $\cos(35) = -0.903$, a negative number where a positive one belongs, which is at least loud enough to notice. The third is a domain limit that is really a geometry lesson: $a/h$ can never exceed 1, because a leg cannot outrun the hypotenuse, so an arccos that refuses to evaluate is telling you the two lengths do not form a right triangle. Finally, solving for the hypotenuse divides by $\cos\theta$, which collapses toward zero as the angle nears 90°: near-vertical geometry makes this rearrangement extremely sensitive to a small error in the angle.