Analytic Geometry and Trigonometry

FE Exam Weight: Mathematics accounts for 8–12 questions (~9% of the 110-question FE Other Disciplines exam). Analytic geometry and trigonometry underpin statics, dynamics, and surveying problems, so the payoff extends well beyond this section.

How the Open-Resource Format Changes Your Strategy

The FE is open to the searchable electronic NCEES FE Reference Handbook only. Every formula in this section — distance, conic standard forms, the Law of Cosines, trig identities — is printed there under Mathematics → Analytic Geometry. The exam therefore does not reward memorizing formulas you can look up; it rewards (1) recognizing which formula applies, (2) plugging numbers in without algebra slips, and (3) doing it fast enough to clear ~110 questions in roughly 5 hours 20 minutes (under 3 minutes each). Treat the Handbook as a tool you must navigate, not a crutch.

Coordinate Systems

Rectangular (Cartesian) points are (x, y) in 2-D or (x, y, z) in 3-D. Polar points (r, θ) give distance r from the origin and angle θ from the +x-axis.

Distance between (x₁, y₁) and (x₂, y₂): d = √((x₂−x₁)² + (y₂−y₁)²). Midpoint: M = ((x₁+x₂)/2, (y₁+y₂)/2).

Worked example — distance. Find the distance from (3, −1) to (−2, 4): d = √((−2−3)² + (4−(−1))²) = √((−5)² + (5)²) = √(25+25) = √50 = 5√2 ≈ 7.07. The most common trap is mishandling the double negative in (4 − (−1)) = 5.

Lines and Slopes

Parallel lines share slope (m₁ = m₂). Perpendicular lines obey m₁·m₂ = −1. The distance from point (x₀, y₀) to line Ax + By + C = 0 is d = |Ax₀ + By₀ + C| / √(A² + B²).

Worked example — point-to-line. Distance from the origin to 3x + 4y = 12: first rewrite as 3x + 4y − 12 = 0 so C = −12. Then d = |3(0) + 4(0) − 12| / √(3² + 4²) = 12/5 = 2.4. The trap: forgetting to move 12 to the left so the formula's sign convention applies.

Conic Sections

All four conics are special cases of slicing a cone, and the Handbook lists each standard form with center (h, k).

Eccentricity (e) is the single number that classifies a conic and frequently appears as a stand-alone FE question:

For an ellipse, c² = a² − b² and e = c/a; for a hyperbola, c² = a² + b² and e = c/a. The asymptotes of a hyperbola are y − k = ±(b/a)(x − h).

Worked example — identify the conic. Given 4x² + 9y² − 36 = 0, divide by 36: x²/9 + y²/4 = 1. Both squared terms are positive with unequal denominators → ellipse with a = 3, b = 2. Then c = √(9−4) = √5, so e = √5/3 ≈ 0.745, confirming 0 < e < 1.

Trigonometry — The Unit Circle

Fundamental Identities

• Pythagorean: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = csc²θ
• Double-angle: sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ
• Sum/difference: sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B

Solving Triangles

For any triangle with sides a, b, c opposite angles A, B, C:

• Law of Sines: a/sin A = b/sin B = c/sin C
• Law of Cosines: c² = a² + b² − 2ab cos C

Worked example — Law of Cosines. A triangle has a = 5, b = 7, included angle C = 60°. Find c: c² = 5² + 7² − 2(5)(7)cos 60° = 25 + 49 − 70(0.5) = 74 − 35 = 39, so c = √39 ≈ 6.24. Because the angle is between the two known sides (SAS), the Law of Cosines is the only direct route — using the Law of Sines here would leave two unknowns.