Tier 1 · act

ACT Trig Deep Dive

Extended trig reference for the ACT — unit circle values, identities, inverse-trig, and the angle-reference trick.

Files (1)

ACTTRIG.py1129 bytes

def p(s):print(s)
def w():input("...")
def o1():
 p("=UNIT CIRCLE=")
 p("0 deg:(1,0)")
 p("30:(rt3/2,")
 p("1/2)")
 w()
 p("45:(rt2/2,")
 p("rt2/2)")
 p("60:(1/2,")
 p("rt3/2)")
 w()
 p("90 deg:(0,1)")
 p("x=cos, y=sin")
 w()
def o2():
 p("=PYTH ID=")
 p("sin^2 x+")
 p("cos^2 x=1")
 w()
 p("tan^2 x+1=")
 p("sec^2 x")
 p("1+cot^2 x=")
 p("csc^2 x")
 w()
def o3():
 p("=DOUBLE ANG=")
 p("sin(2x)=")
 p("2*sin x")
 p("*cos x")
 w()
 p("cos(2x)=")
 p("cos^2 x-")
 p("sin^2 x")
 p("=1-2sin^2 x")
 p("=2cos^2 x-1")
 w()
def o4():
 p("=INVERSE=")
 p("if sin x=k:")
 p("x=arcsin k")
 p("range -90 to")
 p("90 deg")
 w()
 p("if cos x=k:")
 p("x=arccos k")
 p("range 0 to")
 p("180 deg")
 w()
def o5():
 p("=REF ANGLE=")
 p("Q1: ref=x")
 p("Q2: 180-x")
 p("Q3: x-180")
 p("Q4: 360-x")
 w()
 p("Signs: ASTC")
 p("All/Sin/Tan/Cos")
 p("by quadrant")
 w()
def menu():
 while True:
  p("=ACT TRIG=")
  p("1:UNIT CIRC")
  p("2:PYTH ID")
  p("3:DOUBLE ANG")
  p("4:INVERSE")
  p("5:REF ANGLE")
  p("0:EXIT")
  c=input("?>")
  if c=="1":o1()
  elif c=="2":o2()
  elif c=="3":o3()
  elif c=="4":o4()
  elif c=="5":o5()
  elif c=="0":break
menu()

How to use it

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.

Launch ACTTRIG. Pick the topic matching your question:

1:UNIT CIRC — key angle (x, y) values at 0°, 30°, 45°, 60°, 90°
2:PYTH ID — the three Pythagorean identities
3:DOUBLE ANG — sin(2x) and three forms of cos(2x)
4:INVERSE — arcsin, arccos definitions and ranges
5:REF ANGLE — how to compute reference angles by quadrant + the ASTC mnemonic for sign

Example problem

ACT: What is sin(150°)?

150° is in Q2. Need the reference angle + sign.

Run ACTTRIG, pick 5:REF ANGLE.
Screen shows: Q2: 180-x, so ref = 180 − 150 = 30°.
ASTC: Q2 has Sin positive, so sin(150°) = +sin(30°).
Back to 1:UNIT CIRC, 30°: (rt3/2, 1/2) — sin is the y-coord, so sin(30°) = 1/2.
Answer: sin(150°) = 1/2. ✓

TI-84 Plus CE — try before committingoff

TI-84 Plus CE




  (calculator off)

  Press ON to start.

Test cases (2)

ID	Inputs	Expected contains
unit-circle	1, , , , 0	0 deg:(1,0), 45:(rt2/2,
ref-angle	5, , , 0	Q1: ref=x, ASTC