L2 · Sine, cosine, tangent: finding a side
“I can name the three ratios - sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent - choose the ratio that matches the sides I know and want, and rearrange the equation to isolate the unknown side, including when the unknown sits in the denominator (the A1 move).”
A flagpole cable is anchored 20 m of cable (hypotenuse) long, making a 35° angle with the ground. How high up the pole is the cable attached? A calculator gives sin 35° ≈ 0.574. Step 1 (label from the angle): marked angle is at the ground. Known: hypotenuse 20 m. Wanted: the side across from the angle - the height - which is the OPPOSITE side. Step 2 (choose): opposite + hypotenuse = SINE. (Memory hook: SOH CAH TOA - Sin = Opp/Hyp, Cos = Adj/Hyp, Tan = Opp/Adj.) Step 3 (set up and solve): sin 35° = height/20, so height = 20 x 0.574 = 11.48, about 11.5 m. Step 4 (reasonableness): the height must be less than the 20 m cable, and more than half of it (sin 35° > 0.5). Say it plainly: choosing the ratio is the skill; the calculator supplies its value. Wrong ratio = wrong answer with perfect arithmetic.
You know the adjacent side and the hypotenuse and want nothing else. The ratio that connects exactly those two sides is...
A ladder 10 m long leans at 60° to the ground. cos 60° = 0.5. How far is its foot from the wall? -> 5 m (adjacent = 10 x 0.5)
"sin 48° ≈ 0.743. Hypotenuse 12 m. Opposite = 12 x ___ = ___ m." -> 0.743; 8.916, about 8.9 m
10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.