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Analytic Geometry and Trigonometry

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FE Fundamentals of Engineering Exam

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1 Introduction to the FE Exam

About the FE Exam FE Exam Study Strategies Navigating the FE Reference Handbook

2 Mathematics (8–12 Questions)

Analytic Geometry and Trigonometry Single-Variable Calculus Differential Equations Linear Algebra and Matrix Operations Numerical Methods

3 Probability and Statistics (6–9 Questions)

Probability Fundamentals Probability Distributions Statistical Estimation and Hypothesis Testing

4 Chemistry (5–8 Questions)

Chemical Reactions and Stoichiometry Acids, Bases, and pH Oxidation-Reduction and Corrosion

5 Ethics and Professional Practice (5–8 Questions)

Engineering Ethics Fundamentals

6 Engineering Economics (6–9 Questions)

Time Value of Money Economic Analysis Methods

7 Statics (9–14 Questions)

Force Systems and Equilibrium Trusses, Frames, and Machines Centroids and Moments of Inertia Static Friction

8 Dynamics (9–14 Questions)

Particle Kinematics Kinetics: Force, Mass, and Acceleration Work, Energy, Impulse, and Momentum Rigid-Body Rotation and Mechanical Vibrations

9 Strength of Materials (9–14 Questions)

Stress and Strain Fundamentals Beam Analysis: Shear and Moment Diagrams Bending Stress and Torsion Stress Transformation and Material Failure

10 Materials Science (6–9 Questions)

Material Properties and Selection Phase Diagrams and Alloy Systems

11 Fluid Mechanics (12–18 Questions)

Fluid Properties and Statics Fluid Dynamics and Bernoulli's Equation Pumps, Turbines, and Flow Measurement

12 Thermodynamics and Heat Transfer (9–14 Questions)

Laws of Thermodynamics and Properties Thermodynamic Cycles and Applications Heat Transfer: Conduction, Convection, and Radiation

13 Basic Electrical Engineering (6–9 Questions)

Electrical Fundamentals and DC Circuits AC Circuits and Three-Phase Power

14 Safety, Health, and Environment (6–9 Questions)

Industrial Safety and Hazard Management

15 Instrumentation and Controls (4–6 Questions)

Sensors, Data Acquisition, and Control Systems

16 Computational Tools and Computer Science

Spreadsheets, Structured Programming, and Flowcharts

Analytic Geometry and Trigonometry

Key Takeaways Explain This Topic AI

The distance formula d = √((x₂−x₁)² + (y₂−y₁)²) and point-to-line distance d = |Ax₀+By₀+C|/√(A²+B²) are high-frequency FE items.

Eccentricity classifies conics: circle e = 0, ellipse 0 < e < 1, parabola e = 1, hyperbola e > 1.

Memorize the unit-circle values for 0°, 30°, 45°, 60°, 90° — they appear in problems where a calculator is slow.

Law of Cosines handles SAS and SSS triangles; Law of Sines handles AAS/ASA and the ambiguous SSA case.

Every formula here lives in the Analytic Geometry pages of the NCEES FE Reference Handbook — practice FINDING and APPLYING them quickly.

Perpendicular lines have slopes that are negative reciprocals: m₁·m₂ = −1.

Last updated: June 2026

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.

  • Polar → Rectangular Rectangular → Polar
  • x = r cos θ r = √(x² + y²)
  • y = r sin θ θ = arctan(y/x)

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

  • Form Equation Notes
  • Slope-intercept y = mx + b m = slope, b = y-intercept
  • Point-slope y − y₁ = m(x − x₁) through (x₁, y₁)
  • Standard Ax + By = C slope = −A/B

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).

  • Conic Standard form Identifier
  • Circle (x−h)² + (y−k)² = r² x² and y² coefficients equal, same sign
  • Ellipse (x−h)²/a² + (y−k)²/b² = 1 x², y² coefficients unequal, same sign
  • Parabola (x−h)² = 4p(y−k) only ONE squared term
  • Hyperbola (x−h)²/a² − (y−k)²/b² = 1 x², y² coefficients opposite signs

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

  • Conic Eccentricity
  • Circle e = 0
  • Ellipse 0 < e < 1
  • Parabola e = 1
  • Hyperbola e > 1

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

  • θ (°) θ (rad) sin θ cos θ tan θ
  • 0° 0 0 1 0
  • 30° π/6 1/2 √3/2 √3/3
  • 45° π/4 √2/2 √2/2 1
  • 60° π/3 √3/2 1/2 √3
  • 90° π/2 1 0 undefined
  • 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
  • Known Use
  • Two angles + a side (AAS/ASA) Law of Sines
  • Two sides + opposite angle (SSA) Law of Sines (watch ambiguous case)
  • Two sides + included angle (SAS) Law of Cosines
  • Three sides (SSS) Law of Cosines

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.