L1 · Primary ratios from similar right triangles
“I can explain why, in any right triangle with the same acute angle, the ratios opposite/hypotenuse, adjacent/hypotenuse and opposite/adjacent are always the same number — and I can name the hypotenuse, opposite and adjacent sides for a marked angle and compute sin, cos and tan from side lengths.”
Draw a right triangle with acute angle θ = the marked angle. Its sides are: opposite θ = 3, adjacent to θ = 4, hypotenuse = 5. Now double every side: 6, 8, 10. Now triple: 9, 12, 15. Step 1: opposite ÷ hypotenuse: 3/5 = 0.6; 6/10 = 0.6; 9/15 = 0.6. It never moves. Step 2: adjacent ÷ hypotenuse: 4/5 = 8/10 = 12/15 = 0.8. Opposite ÷ adjacent: 3/4 = 6/8 = 9/12 = 0.75. Also constant. Say it plainly: similar right triangles share angles, so these three ratios are properties of the angle, not of the triangle. That is the whole idea of trigonometry — measure the ratio once, use it at any size.
A fourth similar triangle has hypotenuse 25. Without measuring, opposite ÷ hypotenuse is still 0.6 because…
A similar triangle has hypotenuse 20 and the same marked angle θ. Using the ratio 0.6, find the side opposite θ -> 12 (20 × 0.6 = 12)
"Same family, hypotenuse 50: opposite = 50 × ___ = ___" -> 0.6; 30
10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.