20 high-difficulty questions — the topics that sink most students
Trigonometry questions on the ACT test sine, cosine, tangent ratios, special triangles, the unit circle, and identities. These appear in ~4–5 questions and are among the most missed.
Special angles: sin30°=½, cos30°=√3/2, sin45°=√2/2, cos45°=√2/2, sin60°=√3/2, cos60°=½
CAST rule: In Q1 all positive, Q2 sin+, Q3 tan+, Q4 cos+
If sinθ = 3/5 and θ is in Q2, find cosθ.
Solution: cos²θ = 1 − sin²θ = 1 − 9/25 = 16/25 → cosθ = −4/5 (negative in Q2) ✓
Complex numbers involve i = √(−1). The ACT tests arithmetic with complex numbers and powers of i.
To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator. Powers of i cycle with period 4: find remainder when exponent ÷ 4.
Simplify: (3+2i)(1−i)
Solution: = 3−3i+2i−2i² = 3−i−2(−1) = 3−i+2 = 5−i ✓
ACT tests basic matrix operations: addition, scalar multiplication, and 2×2 matrix multiplication. Determinants may also appear.
Matrix multiplication is NOT commutative (AB ≠ BA in general). The product of an m×n matrix and an n×p matrix is an m×p matrix. Only multiply if inner dimensions match.
det([3 2; 1 4]) = 3·4 − 2·1 = 12 − 2 = 10 ✓
Log and exponential questions test properties, change of base, and solving equations. These are consistently among the most missed ACT questions.
Always convert log equations to exponential form: log₂(8) = 3 because 2³ = 8. For equations like log(x) + log(x−3) = 1, combine logs first then convert to exponential.
Solve: log₃(x) = 4
Solution: 3⁴ = x → x = 81 ✓
Arithmetic and geometric sequences appear frequently. The ACT also tests sum formulas.
For arithmetic: common difference d = a₂ − a₁. For geometric: common ratio r = a₂/a₁. Always identify which type before applying formulas.
Find the 10th term of: 3, 7, 11, 15, ...
Solution: d = 4, a₁₀ = 3 + 9(4) = 3 + 36 = 39 ✓
ACT statistics covers mean, median, mode, standard deviation, and basic probability including combinations and permutations.
Higher standard deviation = more spread out data. Adding the same constant to all values shifts mean but does NOT change SD. Multiplying all values by k multiplies SD by |k|.
How many ways to choose 3 students from 8?
Solution: 8C3 = 8!/(3!·5!) = (8·7·6)/(3·2·1) = 56 ✓
Circles, parabolas, and their equations. Distance, midpoint, and slope formulas also tested.
For circles, complete the square to find center and radius. A circle x²+y²+Dx+Ey+F=0 has center (−D/2, −E/2) and r = √((D/2)²+(E/2)²−F).
Center and radius of (x−2)²+(y+3)²=25?
Solution: Center (2,−3), radius = 5 ✓
Factoring, remainder theorem, rational roots, and simplifying rational expressions.
To find if (x−c) is a factor of f(x), simply evaluate f(c). If f(c) = 0, it IS a factor. This is faster than long division on the ACT.
Is (x−2) a factor of f(x) = x³−3x²+x+2?
Solution: f(2) = 8−12+2+2 = 0 → YES, (x−2) is a factor ✓