Pre-Calculus
"Future Engineer/Scientist! Conic Sections at Trigonometry - mga foundation topics para sa Calculus at Engineering. Wag mahilo sa ikot, master mo ang patterns!"
1 Introduction to Conic Sections
Conic sections are curves formed when a plane intersects a double cone.
Named "conics" because they come from slicing a cone at different angles.
Circle
Plane perpendicular to axis
Parabola
Plane parallel to slant
Ellipse
Plane at an angle
Hyperbola
Plane parallel to axis
2 Circle - Standard and General Form
Standard Form
(x - h)² + (y - k)² = r²
- Center: (h, k)
- Radius: r
- Example: (x - 2)² + (y + 3)² = 16
- Center: (2, -3), Radius: 4
General Form
x² + y² + Dx + Ey + F = 0
- h = -D/2
- k = -E/2
- r² = h² + k² - F
- Convert by completing the square
3 Parabola - Vertex, Focus, Directrix
| Opens | Standard Form | Focus | Directrix |
|---|---|---|---|
| Up | (x-h)² = 4p(y-k) | (h, k+p) | y = k - p |
| Down | (x-h)² = -4p(y-k) | (h, k-p) | y = k + p |
| Right | (y-k)² = 4p(x-h) | (h+p, k) | x = h - p |
| Left | (y-k)² = -4p(x-h) | (h-p, k) | x = h + p |
Key Terms:
Vertex: (h, k) | Focus: Point inside | Directrix: Line outside | p: Distance from vertex to focus
4 Ellipse - Major and Minor Axis
Horizontal Major Axis (a > b)
(x-h)²/a² + (y-k)²/b² = 1
- Center: (h, k)
- Vertices: (h±a, k)
- Co-vertices: (h, k±b)
- Foci: (h±c, k)
Vertical Major Axis (a > b)
(x-h)²/b² + (y-k)²/a² = 1
- Center: (h, k)
- Vertices: (h, k±a)
- Co-vertices: (h±b, k)
- Foci: (h, k±c)
Important Relationship:
c² = a² - b²
where a = semi-major axis, b = semi-minor axis, c = distance from center to focus
5 Hyperbola - Two Separate Branches
Horizontal Transverse Axis
(x-h)²/a² - (y-k)²/b² = 1
- Opens: Left and Right
- Vertices: (h±a, k)
- Foci: (h±c, k)
- Asymptotes: y-k = ±(b/a)(x-h)
Vertical Transverse Axis
(y-k)²/a² - (x-h)²/b² = 1
- Opens: Up and Down
- Vertices: (h, k±a)
- Foci: (h, k±c)
- Asymptotes: y-k = ±(a/b)(x-h)
Hyperbola Relationship (different from ellipse!):
c² = a² + b²
6 Trigonometric Functions and Identities
SOH-CAH-TOA
sin θ = O/H
Opposite / Hypotenuse
cos θ = A/H
Adjacent / Hypotenuse
tan θ = O/A
Opposite / Adjacent
Reciprocal Functions
- csc θ = 1/sin θ (cosecant)
- sec θ = 1/cos θ (secant)
- cot θ = 1/tan θ (cotangent)
Pythagorean Identities
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = csc²θ
7 The Unit Circle - Must Memorize!
Common Angle Values (First Quadrant)
| Degrees | Radians | 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 |
Quadrant Signs (ASTC)
All Students Take Calculus
- Q1: All positive
- Q2: Sin positive
- Q3: Tan positive
- Q4: Cos positive
Radian Conversion
180° = π radians
- Degrees to Radians: × π/180
- Radians to Degrees: × 180/π
Practice Questions
1. What is the center and radius of (x-3)² + (y+2)² = 25?
Show Answer
Center: (3, -2), Radius: 5. From standard form (x-h)² + (y-k)² = r², we get h=3, k=-2, r=√25=5.
2. For the ellipse x²/16 + y²/9 = 1, what is the value of c?
Show Answer
c = √7. Using c² = a² - b², we get c² = 16 - 9 = 7, so c = √7.
3. What is sin 60°?
Show Answer
sin 60° = √3/2. This is one of the special angles you should memorize from the unit circle.
4. Convert 120° to radians.
Show Answer
2π/3 radians. 120° × (π/180) = 120π/180 = 2π/3.
Exam Tips for Pre-Calculus
- ✓ Memorize the unit circle - 0°, 30°, 45°, 60°, 90° values for sin, cos, tan
- ✓ Ellipse vs Hyperbola: Ellipse uses c²=a²-b², Hyperbola uses c²=a²+b²
- ✓ ASTC mnemonic: All Students Take Calculus for quadrant signs
- ✓ Completing the square: Essential for converting general to standard form
- ✓ Practice graphing: Know the shapes and key features of each conic section
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