Right Triangle Trigonometry

Relative to a chosen angle $\theta$ (not the right angle), the three sides of a right triangle are named: the hypotenuse (always opposite the right angle, the longest side), the opposite side (across from $\theta$), and the adjacent side (next to $\theta$, but not the hypotenuse). Now draw two right triangles that both contain a $37°$ angle, one small and one huge. Their angles match ($90°$, $37°$ and $53°$), so the triangles are similar: one is just a scaled copy of the other. Scaling multiplies every side by the same factor, so the RATIOS between sides are identical. A 3-4-5 triangle and a 6-8-10 triangle with the same angle both give $\frac{\text{opp}}{\text{hyp}}=\frac{3}{5}=\frac{6}{10}=0.6$. In a right triangle, once you fix one of the non-right angles, the ratios between the three sides never change, no matter how big or small the triangle is. Those fixed ratios are called sine, cosine and tangent, and they're the whole engine behind right-triangle trigonometry.

Each primary ratio has an upside-down partner. Cosecant is the reciprocal of sine, secant is the reciprocal of cosine, and cotangent is the reciprocal of tangent: $\csc \theta =\frac{\text{hyp}}{\text{opp}}=\frac{1}{\sin \theta }$, $\sec \theta =\frac{\text{hyp}}{\text{adj}}=\frac{1}{\cos \theta }$ and $\cot \theta =\frac{\text{adj}}{\text{opp}}=\frac{1}{\tan \theta }$. A quick way to pair them: each reciprocal pairs with the primary ratio whose name does NOT start with the same letters (cosecant with sine, secant with cosine). Most calculators have no csc, sec or cot button, so evaluate them as 1 divided by the primary ratio: $\sec 40°=\frac{1}{\cos 40°}\approx \frac{1}{0.7660}\approx 1.305\text{.}$ Because the hypotenuse is the longest side, $\csc \theta$ and $\sec \theta$ are always at least $1$ for an acute angle, a handy check on your answer.

The inverse keys (${\sin }^{-1}$, ${\cos }^{-1}$, ${\tan }^{-1}$) run a ratio backwards: you feed in a ratio and get back the angle that produces it. Always confirm the calculator shows DEG, since MCR3U works entirely in degrees. Once two parts are known, you can finish the rest of the triangle the cheap way: the two acute angles add to $90°$, and the Pythagorean theorem $a^2+b^2=c^2$ gives a third side without any trig. Use it to check: the hypotenuse must come out longest.

An angle of elevation is measured UPWARD from the horizontal (e.g. looking up at the top of a building). An angle of depression is measured DOWNWARD from the horizontal (e.g. looking down at a boat from a cliff). Crucially, the angle of elevation from the ground up to a point equals the angle of depression from that point back down to the ground: the two horizontal lines are parallel and the line of sight cuts across both, so the angles are alternate angles. In practice this means you can always move the given angle to the bottom of the triangle, where the right angle and the ground distance are easy to see.

Angle of Elevation and Depression Word Problems Trigonometry, Finding Sides, Angles, Right Triangles

Exam problems often hide two right triangles in one picture: two boats seen from one lighthouse, a flagpole standing on a building, or two observers on either side of a tower. The strategy is always the same. Find the side the two triangles SHARE (usually a height or a ground distance), solve each triangle separately for the piece you need, then add or subtract. If both boats are on the same side of the lighthouse, the gap between them is a difference of two distances; if the observers stand on opposite sides of the tower, their distances add. Keep full calculator values until the final subtraction, because subtracting two rounded numbers magnifies the rounding error.

Distance Between 2 Ships (Angle of Depression)