SL 3.2—2d and 3d trig

The 2.5 m distance from the wall is adjacent to the 62° angle, and the ladder itself is the hypotenuse.

Since we relate adjacent and hypotenuse, use cosine: $\cos 62°=\frac{2.5}{L}$.

$L=\frac{2.5}{\cos 62°}$.

$L=\frac{2.5}{0.4695}\approx 5.33$ m.

$2.66$ m results from using sine instead of cosine, mismatching the side to the angle.

$4.70$ m comes from multiplying $2.5\times \cos 62°$ instead of dividing.

$1.33$ m comes from computing $2.5\times \tan 62°$ then halving incorrectly, an arithmetic slip.

Only dividing 2.5 by $\cos 62°$ correctly isolates the hypotenuse.

Opposite = 7 cm, adjacent = 4 cm, so use tangent.

$\tan \theta =\frac{7}{4}=1.75$.

$\theta =\arctan (1.75)$.

$\theta \approx 60.3°$.

$29.7°$ is the complement of the correct angle, from swapping opposite and adjacent.

$34.9°$ comes from computing $\arctan (4/7)$, again swapping the ratio.

$55.1°$ results from mistakenly using $\arcsin (7/8.06)$ with an incorrect hypotenuse estimate.

Only $\arctan (7/4)$ correctly uses opposite over adjacent.