SL 3.3—Angles of elevation and depression

The horizontal distance 60 m is adjacent to the angle of elevation, and the height 45 m is opposite.

Since opposite and adjacent are known, use $\tan (\theta )=\frac{45}{60}$.

$\theta ={\tan }^{-1}(0.75)\approx 36.87°$, which rounds to $37°$.

The angle of elevation is approximately $37°$.

$53°$ comes from swapping opposite and adjacent, computing ${\tan }^{-1}(60/45)$ instead.

$41°$ would result from an arithmetic slip in the division before taking inverse tangent.

$49°$ does not correspond to any correct ratio of these two sides.

Only $37°$ correctly uses ${\tan }^{-1}(45/60)$.

The angle of depression equals the angle of elevation from the boat, so use $12°$ in the right triangle with height 80 m opposite and distance $d$ adjacent.

$\tan (12°)=\frac{80}{d}$.

$d=\frac{80}{\tan (12°)}\approx \frac{80}{0.2126}\approx 376.3$ m.

To 3 significant figures, $d\approx 376$ m.

$17.0$ m results from computing $80\times \tan (12°)$ instead of dividing.

$81.8$ m results from using $\cos (12°)$ incorrectly with the hypotenuse assumption.

$82.0$ m comes from mistakenly using $\sin (12°)$ in a hypotenuse calculation.

Only dividing 80 by $\tan (12°)$ gives the correct horizontal distance of $376$ m.