L4 · Solving a complete right triangle
“I can solve a right triangle — find every side and every angle — from any two given pieces (a side and an acute angle, or two sides), choosing Pythagoras, a ratio, an inverse ratio or the 90° complement in an efficient order, and checking the finished triangle against the angle sum and Pythagoras.”
A right triangle has hypotenuse 18 cm and one acute angle of 35°. Solve the triangle. Step 1 (the free angle): the other acute angle is 90° - 35° = 55°. Take it first — it costs nothing. Step 2 (opposite the 35° angle): sin 35° = x/18 -> x = 18 × sin 35° ≈ 18 × 0.574 = 10.3 cm. Step 3 (adjacent to the 35° angle): cos 35° = y/18 -> y = 18 × cos 35° ≈ 18 × 0.819 = 14.7 cm. Step 4 (audit): 10.3² + 14.7² = 106.1 + 216.1 = 322.2 ≈ 18² = 324 ✓ (close — the gap is rounding). Angles: 35 + 55 + 90 = 180 ✓.
Why take the complementary angle first?
Marked angle 62°, adjacent side 7 cm. Find the other angle, the hypotenuse and the opposite side. -> Other angle: 90 - 62 = 28°. Hypotenuse: cos 62° = 7/h -> h = 7 ÷ cos 62° ≈ 14.9 cm. Opposite: tan 62° = x/7 -> x = 7 × tan 62° ≈ 13.2 cm.
"Hypotenuse 30 cm, one leg 18 cm. Other leg = √(30² - 18²) = ___ cm. Angle opposite the 18 cm leg: sin⁻¹(18/30) = ___°. Other acute angle = ___°." -> 24 cm; 37°; 53°
10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.