SMA 2171: GEOMETRY

Course Content

• Trigonometry: Trigonometric functions, graphs, and inverses; identities and equations; sine and cosine formulae.

• Straight Line: Slope, inclination, distance formula, parallel and perpendicular lines.

• Circle: Equations, distance from a point to a circle, tangent and normal lines, orthogonal circles.

• Ellipse, Parabola, Hyperbola: Standard forms, chords, tangents, normals.

• Polar Coordinates: Graphs and equations.

• Vectors: Operations in two/three dimensions, applications to trigonometry and geometry.

1. Trigonometry

1.1 Circular Functions and Angle Measures

• Angle Measurement: Rotation of line OA relative to the origin.

  • Conversion: $180^\circ = \pi \text{ radians}$

  • Arc Length: $s = r \theta$ ($\theta$ in radians)

  • Sector Area: $A = \frac{1}{2} r^2 \theta$

1.2 Trigonometric Ratios and Identities

• SOHCAHTOA:

  • $\text{sin } \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$

  • $\text{cos } \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$

  • $\text{tan } \theta = \frac{\text{Opposite}}{\text{Adjacent}}$

• Fundamental Identities:

  • Pythagorean: $\text{sin}^2\theta + \text{cos}^2\theta = 1$, $1 + \text{tan}^2\theta = \text{sec}^2\theta$, $1 + \text{cot}^2\theta = \text{csc}^2\theta$

  • Double Angle:

  • $\text{sin}(2\theta) = 2 \text{sin} \theta \text{cos} \theta$

  • $\text{cos}(2\theta) = \text{cos}^2\theta - \text{sin}^2\theta = 2\text{cos}^2\theta - 1 = 1 - 2\text{sin}^2\theta$

1.3 Laws of Triangles

• Sine Rule: $\frac{a}{\text{sin } A} = \frac{b}{\text{sin } B} = \frac{c}{\text{sin } C}$

• Cosine Rule: $a^2 = b^2 + c^2 - 2bc\text{ cos } A$

2. Straight Lines

2.1 Slope and Distance

• Distance Formula: Between $A(x<em>1, y</em>1)$ and $B(x<em>2, y</em>2)$: $d = \sqrt{(x<em>2 - x</em>1)^2 + (y<em>2 - y</em>1)^2}$

• Slope ($m$): $m = \text{tan } \theta = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$

2.2 Line Equations

• Slope-Intercept Form: $y = mx + c$

• Point-Slope Form: $y - y<em>1 = m(x - x</em>1)$

• General Form: $Ax + By + C = 0$

2.3 Geometry of Lines

• Parallel: $m<em>1 = m</em>2$

• Perpendicular: $m<em>1 \cdot m</em>2 = -1$

3. Conic Sections

3.1 Circle

• Standard Form: $(x-h)^2 + (y-k)^2 = r^2$

• General Equation: $x^2 + y^2 + 2gx + 2fy + c = 0$, where center is $(-g, -f)$ and radius $r = \sqrt{g^2 + f^2 - c}$.

3.2 Parabola

• Standard Forms (vertex at origin):

  • $y^2 = 4ax$: Opens right, focus $(a, 0)$, directrix $x = -a$.

  • $x^2 = 4ay$: Opens up, focus $(0, a)$, directrix $y = -a$.

3.3 Ellipse

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

• Eccentricity ($e$): $e = \frac{c}{a}$ where $c^2 = a^2 - b^2$.

3.4 Hyperbola

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

• Asymptotes: $y - k = \pm \frac{b}{a}(x - h)$

4. Polar Coordinates

• Cartesian to Polar: $r^2 = x^2 + y^2$, $\text{tan } \theta = \frac{y}{x}$

• Polar to Cartesian: $x = r \cos \theta$, $y = r \sin \theta$

5. Vectors

• Magnitude: $|\mathbf{v}| = \sqrt{v<em>x^2 + v</em>y^2}$

• Dot Product: $\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}| \cos \theta = a<em>x b</em>x + a<em>y b</em>y$

• Vector Form: $\mathbf{v} = v<em>x \mathbf{i} + v</em>y \mathbf{j}$