Comprehensive Math Reviewer: Cartesian Transformations and Trigonometric Laws

MATH REVIEWER : TRANSLATION IN THE CARTESIAN PLANE ( SORRY GUYS DAGHAN KAAYOG TYPO )

• Definition of Translation: A transformation that slides a point or a figure across the Cartesian plane to a new location without changing its size, shape, or orientation.

• Positional Coordinate Adjustments:

  • Rightward Movement: If a point moves to the right, the $x$ coordinates increase.

  • Leftward Movement: If a point moves to the left, the $x$ coordinates decrease.

  • Upward Movement: If a point moves up, the $y$ coordinates increase.

  • Downward Movement: If a point moves down, the $y$ coordinates decrease.

• Specific Example of Translation:

  • Initial point: $B = (-2, 4)$

  • Transformation calculation provided: $(-2 - 2, 4 - 3)$

  • Resulting point: $B'(-4, 1)$

REFLECTION

• Definition of Reflection: A transformation that flips a point or figure across a specific line known as the line of reflection. Each point of the reflected image is the same distance from the line of reflection as the corresponding point of the original figure.

• General Coordinate Rules for Reflection:

  • Across the X-axis: $(x, -y)$

  • Across the Y-axis: $(-x, y)$

  • Across the Diagonal line $y = x$: $(y, x)$

  • Across the Diagonal line $y = -x$: $(y, -x)$

• Reflection Data Table (as provided in transcript notes):

ROTATIONS

• Definition of Rotation: The transformation of an object by turning it about a fixed point (the center of rotation) without altering the shape or the size of the object.

• Types of Rotation and Coordinate Rules:

LAW OF SINES

• Fundamental Formula:   $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$

• Ambiguous Case Analysis: Occurs when two sides and a non-included angle (SSA) are given. The following conditions determine the existence and quantity of triangles:

  • Case 1: No such triangle exists: If $\angle A$ is an acute angle and a < h (where $h$ is the altitude/height).

  • Case 2: Two different triangles exist: If $\angle A$ is acute and h < a < b.

  • Case 3: Exactly one triangle exists: If $a = h$, then the triangle formed is a right triangle.

LAW OF COSINES

• Calculating a Missing Side:

  • If side $a$ is missing: $a^2 = b^2 + c^2 - 2bc \cos(A)$

  • If side $b$ is missing: $b^2 = a^2 + c^2 - 2ac \cos(B)$

  • If side $c$ is missing: $c^2 = a^2 + b^2 - 2ab \cos(C)$

• Calculating a Missing Angle:

  • $A = \cos^{-1} \left( \frac{b^2 + c^2 - a^2}{2bc} \right)$

  • Note: OVER NA MGA HAJIMA

  • $B = \cos^{-1} \left( \frac{a^2 + c^2 - b^2}{2ac} \right)$

  • $C = \cos^{-1} \left( \frac{a^2 + b^2 - c^2}{2ab} \right)$

Questions & Discussion

• General Note found in transcript: "SORRY GUYS DAGHAN KAAYOG TYPO" - this serves as a disclaimer regarding potential typographical errors in the provided reviewer material.

• Informal Interjection: "OVER NA MGA HAJIMA" - included alongside the law of cosines angle formulas.