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.

Conic Sections and the Law of Cosines

Beyond lines and triangles, the FE expects recognition of the conic sections from their standard equations. A circle is (x−h)²+(y−k)²=r²; an ellipse is x²/a²+y²/b²=1; a parabola is y=ax²+bx+c; and a hyperbola is x²/a²−y²/b²=1. Identifying the conic from its equation — and reading off center, vertex, or radius — is a common quick point.

For oblique triangles (no right angle), use the law of cosines c²=a²+b²−2ab·cos C and the law of sines a/sin A = b/sin B = c/sin C.

Worked Example

Two sides of a triangle are 5 and 7 with a 60° included angle. The third side is c = √(25+49−2·5·7·cos60°) = √(74−70·0.5) = √39 ≈ 6.24. Watch the ambiguous case of the law of sines, where two triangles may satisfy the given data.