{"slug":"act-trig-pack","tier":1,"subject":"act","title":"ACT Trig Deep Dive","description":"Extended trig reference for the ACT — unit circle values, identities, inverse-trig, and the angle-reference trick.","files":[{"filename":"ACTTRIG.py","content":"def p(s):print(s)\ndef w():input(\"...\")\ndef o1():\n p(\"=UNIT CIRCLE=\")\n p(\"0 deg:(1,0)\")\n p(\"30:(rt3/2,\")\n p(\"1/2)\")\n w()\n p(\"45:(rt2/2,\")\n p(\"rt2/2)\")\n p(\"60:(1/2,\")\n p(\"rt3/2)\")\n w()\n p(\"90 deg:(0,1)\")\n p(\"x=cos, y=sin\")\n w()\ndef o2():\n p(\"=PYTH ID=\")\n p(\"sin^2 x+\")\n p(\"cos^2 x=1\")\n w()\n p(\"tan^2 x+1=\")\n p(\"sec^2 x\")\n p(\"1+cot^2 x=\")\n p(\"csc^2 x\")\n w()\ndef o3():\n p(\"=DOUBLE ANG=\")\n p(\"sin(2x)=\")\n p(\"2*sin x\")\n p(\"*cos x\")\n w()\n p(\"cos(2x)=\")\n p(\"cos^2 x-\")\n p(\"sin^2 x\")\n p(\"=1-2sin^2 x\")\n p(\"=2cos^2 x-1\")\n w()\ndef o4():\n p(\"=INVERSE=\")\n p(\"if sin x=k:\")\n p(\"x=arcsin k\")\n p(\"range -90 to\")\n p(\"90 deg\")\n w()\n p(\"if cos x=k:\")\n p(\"x=arccos k\")\n p(\"range 0 to\")\n p(\"180 deg\")\n w()\ndef o5():\n p(\"=REF ANGLE=\")\n p(\"Q1: ref=x\")\n p(\"Q2: 180-x\")\n p(\"Q3: x-180\")\n p(\"Q4: 360-x\")\n w()\n p(\"Signs: ASTC\")\n p(\"All/Sin/Tan/Cos\")\n p(\"by quadrant\")\n w()\ndef menu():\n while True:\n  p(\"=ACT TRIG=\")\n  p(\"1:UNIT CIRC\")\n  p(\"2:PYTH ID\")\n  p(\"3:DOUBLE ANG\")\n  p(\"4:INVERSE\")\n  p(\"5:REF ANGLE\")\n  p(\"0:EXIT\")\n  c=input(\"?>\")\n  if c==\"1\":o1()\n  elif c==\"2\":o2()\n  elif c==\"3\":o3()\n  elif c==\"4\":o4()\n  elif c==\"5\":o5()\n  elif c==\"0\":break\nmenu()"}],"testCases":[{"id":"unit-circle","inputs":["1","","","","0"],"expectedContains":["0 deg:(1,0)","45:(rt2/2,"]},{"id":"ref-angle","inputs":["5","","","0"],"expectedContains":["Q1: ref=x","ASTC"]}],"usage":"The ACT asks about 3-4 trig questions per test and they hit topics the formula sheet doesn't cover — unit circle values, identities, inverse functions, quadrant sign rules. This program is the cheat-sheet companion to the main ACT formula pack.\n\nLaunch `ACTTRIG`. Pick the topic matching your question:\n\n- `1:UNIT CIRC` — key angle (x, y) values at 0°, 30°, 45°, 60°, 90°\n- `2:PYTH ID` — the three Pythagorean identities\n- `3:DOUBLE ANG` — sin(2x) and three forms of cos(2x)\n- `4:INVERSE` — arcsin, arccos definitions and ranges\n- `5:REF ANGLE` — how to compute reference angles by quadrant + the ASTC mnemonic for sign","example":{"problem":"ACT: What is sin(150°)?","walkthrough":"150° is in Q2. Need the reference angle + sign.\n\n1. Run `ACTTRIG`, pick `5:REF ANGLE`.\n2. Screen shows: `Q2: 180-x`, so ref = 180 − 150 = **30°**.\n3. ASTC: Q2 has Sin positive, so sin(150°) = +sin(30°).\n4. Back to `1:UNIT CIRC`, 30°: (rt3/2, 1/2) — sin is the y-coord, so sin(30°) = **1/2**.\n5. Answer: sin(150°) = **1/2**. ✓"}}