Pre-Calculus
Conic Sections, Trigonometric Identities & Circular Functions
In This Lesson
Conic Sections
Conic sections are curves formed by the intersection of a plane with a double-napped cone. The four types are: circle, parabola, ellipse, and hyperbola.
Circle
(x - h)² + (y - k)² = r²
- Center: (h, k)
- Radius: r
- All points are equidistant from the center
Parabola
y = a(x - h)² + k
- Vertex: (h, k)
- Opens up: a > 0
- Opens down: a < 0
Ellipse
(x-h)²/a² + (y-k)²/b² = 1
- Center: (h, k)
- Major axis: 2a (longer)
- Minor axis: 2b (shorter)
Hyperbola
(x-h)²/a² - (y-k)²/b² = 1
- Center: (h, k)
- Transverse axis: 2a
- Asymptotes: y = ±(b/a)x
General Second-Degree Equation
Ax² + Bxy + Cy² + Dx + Ey + F = 0
Use B² - 4AC to identify: Circle (B²-4AC < 0, A=C), Ellipse (B²-4AC < 0), Parabola (B²-4AC = 0), Hyperbola (B²-4AC > 0)
Trigonometric Identities
Pythagorean Identities
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = csc²θ
Reciprocal Identities
cscθ = 1/sinθ
secθ = 1/cosθ
cotθ = 1/tanθ
Double Angle Formulas
- sin(2θ) = 2sinθcosθ
- cos(2θ) = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ
- tan(2θ) = 2tanθ / (1 - tan²θ)
Sum & Difference Formulas
- sin(A ± B) = sinAcosB ± cosAsinB
- cos(A ± B) = cosAcosB ∓ sinAsinB
- tan(A ± B) = (tanA ± tanB) / (1 ∓ tanAtanB)
Circular Functions
Unit Circle
A circle with radius 1 centered at origin. For angle θ, the point (cosθ, sinθ) lies on the circle.
| Angle (°) | Radians | sin | cos | tan |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | 1/√3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | undefined |
Converting Degrees and Radians
Degrees to Radians: θ × (π/180)
Radians to Degrees: θ × (180/π)
Series & Sequences
Arithmetic Sequence
A sequence with a constant difference (d) between terms.
nth term: aₙ = a₁ + (n-1)d
Sum: Sₙ = n/2 × (a₁ + aₙ) = n/2 × [2a₁ + (n-1)d]
Geometric Sequence
A sequence with a constant ratio (r) between terms.
nth term: aₙ = a₁ × r^(n-1)
Sum (r ≠ 1): Sₙ = a₁(1 - rⁿ)/(1 - r)
Infinite Sum (|r| < 1): S∞ = a₁/(1 - r)
Sigma Notation
Compact way to write sums.