ch1 et ch2 math

Expanded Essentials with Details and Formulas

Geometry Essentials

1. Euclidean Geometry

  • Geometry focused on shapes like points, lines, angles, and surfaces.

  • Governed by Euclid’s Five Postulates:

    1. A straight line can be drawn between any two points.

    2. A straight line can be extended infinitely.

    3. A circle can be drawn with any center and radius.

    4. All right angles are congruent.

    5. Through a point not on a line, only one parallel line can be drawn to the given line.

2. Thales' Theorem

  • Theorem: If a line is parallel to one side of a triangle and intersects the other two sides, it divides those sides proportionally.

  • Proportionality Formula:

    AMAB=ANAC=MNBC\frac{AM}{AB} = \frac{AN}{AC} = \frac{MN}{BC}

3. Pythagoras' Theorem

  • Theorem: In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

  • Formula:

    AC2=AB2+BC2AC^2 = AB^2 + BC^2

  • Application: To check if a triangle is a right triangle:

    AC2=?AB2+BC2AC^2 \stackrel{?}{=} AB^2 + BC^2

4. Quadrilaterals and Their Properties

  • Parallelogram:

    • Opposite sides are equal and parallel.

    • Formula: Area =Base×Height= \text{Base} \times \text{Height}.

  • Rectangle:

    • Opposite sides are equal; all angles =90∘= 90^\circ.

    • Formula: Area =Length×Width= \text{Length} \times \text{Width}.

  • Rhombus:

    • All sides equal; diagonals bisect at 90∘90^\circ.

    • Formula: Area =12×Diagonal 1×Diagonal 2= \frac{1}{2} \times \text{Diagonal 1} \times \text{Diagonal 2}.

  • Square:

    • All sides equal; all angles =90∘= 90^\circ.

    • Formula: Area =Side2= \text{Side}^2.

5. Triangles

  • Types:

    • Scalene: No sides equal.

    • Isosceles: Two sides equal.

    • Equilateral: All sides equal.

  • Congruence Criteria:

    • SAS (Side-Angle-Side), ASA (Angle-Side-Angle), SSS (Side-Side-Side).

Trigonometry Essentials

1. Trigonometric Ratios in Right Triangles

  • Sine (sin⁡\sin): sin⁡A=OppositeHypotenuse\sin A = \frac{\text{Opposite}}{\text{Hypotenuse}}

  • Cosine (cos⁡\cos): cos⁡A=AdjacentHypotenuse\cos A = \frac{\text{Adjacent}}{\text{Hypotenuse}}

  • Tangent (tan⁡\tan): tan⁡A=OppositeAdjacent\tan A = \frac{\text{Opposite}}{\text{Adjacent}}

2. Unit Circle

  • Definition: A circle with radius 1 centered at the origin.

  • Coordinates: Any point M(x,y)M(x, y) on the circle represents (cos⁡θ,sin⁡θ)(\cos \theta, \sin \theta).

  • Key Pythagorean Identity:

    sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1

3. Basic Trigonometric Identities

  • Quotient Identities:

    tan⁡(A)=sin⁡(A)cos⁡(A)\tan(A) = \frac{\sin(A)}{\cos(A)}tan(A)=cos(A)sin(A)​cot⁡(A)=cos⁡(A)sin⁡(A)\cot(A) = \frac{\cos(A)}{\sin(A)}cot(A)=sin(A)cos(A)

  • ​Reciprocal Identities:

    csc(A)=sin(A)1​,sin(A)=csc(A)1​ sec⁡(A)=1cos⁡(A),cos⁡(A)=1sec⁡(A)\sec(A) = \frac{1}{\cos(A)}, \quad \cos(A) = \frac{1}{\sec(A)}sec(A)=cos(A)1​,cos(A)=sec(A)1​ cot⁡(A)=1tan⁡(A),tan⁡(A)=1cot⁡(A)\cot(A) = \frac{1}{\tan(A)}, \quad \tan(A) = \frac{1}{\cot(A)}cot(A)=tan(A)1​,tan(A)=cot(A)1​

  • Pythagorean Identities:

    sin2(A)+cos2(A)=1 1+tan⁡2(A)=sec⁡2(A)1 + \tan^2(A) = \sec^2(A)1+tan2(A)=sec2(A) 1+cot⁡2(A)=csc⁡2(A)1 + \cot^2(A) = \csc^2(A)1+cot2(A)=csc2(A

4. Law of Sines and Law of Cosines

  • Law of Sines: asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

  • Law of Cosines: c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab \cos C

5. Radians and Degrees

  • Conversion: Radians=π180×Degrees,Degrees=180π×Radians\text{Radians} = \frac{\pi}{180} \times \text{Degrees}, \quad \text{Degrees} = \frac{180}{\pi} \times \text{Radians}

Even/Odd Identities

Sine and Cosecant:

sin⁡(−A)=−sin⁡(A),(sine is an odd function)\sin(-A) = -\sin(A), \quad \text{(sine is an odd function)}sin(−A)=−sin(A),(sine is an odd function)csc⁡(−A)=−csc⁡(A),(cosecant is an odd function)\csc(-A) = -\csc(A), \quad \text{(cosecant is an odd function)}csc(−A)=−csc(A),(cosecant is an odd function)

Cosine and Secant:

cos⁡(−A)=cos⁡(A),(cosine is an even function)\cos(-A) = \cos(A), \quad \text{(cosine is an even function)}cos(−A)=cos(A),(cosine is an even function)sec⁡(−A)=sec⁡(A),(secant is an even function)\sec(-A) = \sec(A), \quad \text{(secant is an even function)}sec(−A)=sec(A),(secant is an even function)

Tangent and Cotangent:

tan⁡(−A)=−tan⁡(A),(tangent is an odd function)\tan(-A) = -\tan(A), \quad \text{(tangent is an odd function)}tan(−A)=−tan(A),(tangent is an odd function)cot⁡(−A)=−cot⁡(A),(cotangent is an odd function)\cot(-A) = -\cot(A), \quad \text{(cotangent is an odd function)}cot(−A)=−cot(A),(cotangent is an odd function)



Types of Angles

  1. Acute Angles:

    • Measure between 0° and 90° (not inclusive).

  2. Right Angles:

    • Measure exactly 90°.


  3. Obtuse Angles:

    • Measure between 90° and 180° (not inclusive).


  4. Straight Angles:

    • Measure exactly 180°.