Domain 6 • Trigonometry & Analytical Geometry
Pre-Calculus on ALEKS
Pre-Calculus is the final mastery tier on the ALEKS placement test. Scoring in this domain (score 76+) unlocks direct enrollment into freshman Calculus I (Math 220) for STEM, engineering, computer science, and mathematics majors.
Core Tested Concepts
Unit Circle & Exact ValuesMemorizing exact sine, cosine, and tangent values for standard angles (0, π/6, π/4, π/3, π/2 and quadrant multiples).
Radians, Arc Length & Sector AreaConverting between degrees and radians (θ_deg × π/180), arc length formula s = rθ, sector area A = 1/2 r²θ (θ in radians).
Fundamental Trigonometric IdentitiesPythagorean identities (sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ), reciprocal identities (csc, sec, cot), double angle formulas.
2D Vectors & Dot ProductsComponent form <a, b>, magnitude ||v|| = √(a² + b²), scalar multiplication, and algebraic dot product u · v.
High-Yield Cheat Sheet
Essential Pre-Calculus Formulas
Pythagorean Identity
sin²θ + cos²θ = 1
Arc Length
s = r · θ (radians)
Vector Magnitude
||v|| = √(x² + y²)
Practice Drills
5 Authentic Pre-Calculus Practice Questions
1Practice Problem
Find the exact value of cos(2π/3).
Hint: 2π/3 radians is 120° in Quadrant II, where cosine is negative. Reference angle is 60° (π/3).
Formula: cos(120°) = -cos(60°)
2Practice Problem
If sin(θ) = 3/5 and θ is an acute angle in Quadrant I, find the exact value of tan(θ).
Hint: Use a 3-4-5 right triangle: opposite = 3, hypotenuse = 5, adjacent = 4. tan(θ) = opp / adj.
Formula: tan(θ) = sin(θ) / cos(θ)
3Practice Problem
Convert 5π/6 radians to degrees.
Hint: Multiply radians by (180 / π).
Formula: degrees = radians × (180 / π)
4Practice Problem
Simplify the trigonometric expression: (1 - cos²(x)) / sin(x)
Hint: From the Pythagorean identity sin²(x) + cos²(x) = 1, we know 1 - cos²(x) = sin²(x).
Formula: sin²(x) + cos²(x) = 1
5Practice Problem
Given vector u = <3, -4> and vector v = <2, 5>, calculate their dot product u · v.
Hint: Dot product u · v = u_x · v_x + u_y · v_y.
Formula: u · v = x₁x₂ + y₁y₂