Analytic Geometry and Trigonometry

FE Exam Weight: Mathematics accounts for 8-12 questions (~9% of the exam). Analytic geometry and trigonometry form the foundation for nearly every other engineering topic.

Coordinate Systems

Rectangular (Cartesian) Coordinates

Points are defined by (x, y) in 2D or (x, y, z) in 3D.

Distance between two points: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Midpoint: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

Polar Coordinates

Points are defined by (r, θ) where r is the distance from the origin and θ is the angle from the positive x-axis.

Conversions:

Lines and Slopes

Parallel lines have equal slopes: m₁ = m₂

Perpendicular lines have negative reciprocal slopes: m₁ · m₂ = -1

Distance from point (x₀, y₀) to line Ax + By + C = 0: $d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$

Conic Sections

Circle

$\text{Standard form: } (x - h)^2 + (y - k)^2 = r^2$

• Center: (h, k)
• Radius: r

Ellipse

$\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$

• Center: (h, k)
• Semi-major axis: a (if a > b) or b (if b > a)
• Eccentricity: e = c/a where c² = a² - b² (e < 1 for ellipse)

Parabola

$\text{Vertical: } (x - h)^2 = 4p(y - k)$ $\text{Horizontal: } (y - k)^2 = 4p(x - h)$

• Vertex: (h, k)
• Focus distance: |p| from vertex
• Directrix: distance |p| on opposite side of focus

Hyperbola

$\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$

• Center: (h, k)
• Asymptotes: y - k = ±(b/a)(x - h)
• Eccentricity: e = c/a where c² = a² + b² (e > 1 for hyperbola)

Trigonometric Functions

The Unit Circle (Key Values)

Fundamental Identities

Pythagorean Identities:

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = csc²θ

Double-Angle Identities:

• sin(2θ) = 2 sin θ cos θ
• cos(2θ) = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ
• tan(2θ) = 2tan θ / (1 - tan²θ)

Sum and Difference Identities:

• sin(A ± B) = sin A cos B ± cos A sin B
• cos(A ± B) = cos A cos B ∓ sin A sin B
• tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)

Half-Angle Identities:

• sin(θ/2) = ±√((1 - cos θ)/2)
• cos(θ/2) = ±√((1 + cos θ)/2)

Laws of Sines and Cosines

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

Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Law of Cosines: $c^2 = a^2 + b^2 - 2ab\cos C$

When to use which: