Comprehensive Study Note on Trigonometry, Probability, and Triangle Congruence

Trigonometry word problems primarily utilize the fundamental trigonometric ratios to solve for unknown side lengths or angles in right-angled triangles.
The foundational mnemonic used for these ratios is SOH CAH TOA, defined as: - Sine: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$ - Cosine: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ - Tangent: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$
Angles of Elevation and Depression: - The angle of elevation is the angle formed between the horizontal line of sight and the line of sight looking up at an object. - The angle of depression is the angle formed between the horizontal line of sight and the line of sight looking down at an object. - Due to the properties of parallel lines (specifically alternate interior angles), the angle of elevation from point $A$ to point $B$ is numerically equal to the angle of depression from point $B$ to point $A$.
Procedural Steps for Solving: - 1. Draw a clear diagram based on the verbal description. - 2. Label the known values (angles and side lengths). - 3. Identify the target variable (what the problem is asking to find). - 4. Select the appropriate trigonometric ratio based on the relationship between the knowns and the unknown. - 5. Set up the equation and solve for the variable, ensuring the calculator is in the correct mode (Degrees vs. Radians).

Fundamental Probability Definition: - The probability of an event occurring, denoted as $P(E)$ , is the ratio of the number of favorable outcomes to the total number of possible outcomes in the sample space. - Formula: $P(E) = \frac{n(E)}{n(S)}$ , where $n(E)$ is the number of ways the event can occur and $n(S)$ is the total size of the sample space.
Spinner Problems and Discrete Outcomes: - In multi-spin scenarios, the total number of outcomes is determined by the Fundamental Counting Principle. For example, if a spinner has $n$ equal sections and is spun $k$ times, the total outcomes are $n^k$ . - To calculate the probability of a specific sequence of spins, multiply the individual probabilities of each spin (assuming independence).
Geometric Probability (Area-Based): - Probability can be calculated based on the ratio of lengths, areas, or volumes. - In a 2D plane, if a point is chosen at random within a region $S$, the probability that the point lies within a sub-region $E$ is: - Formula: $P(\text{Area}) = \frac{\text{Area of Favorable Region}}{\text{Total Area of Sample Region}}$ - This application is used for targets, maps, and shaded-region problems.

To prove that two triangles are congruent (identical in shape and size), specific criteria must be met to ensure all corresponding parts are equal without measuring every side and angle.
SSS (Side-Side-Side) Postulate: - If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.
SAS (Side-Angle-Side) Postulate: - If two sides and the included angle (the angle between the two sides) of one triangle are congruent to two sides and the included angle of another triangle, the triangles are congruent.
ASA (Angle-Side-Angle) Postulate: - If two angles and the included side (the side between the two angles) of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent.
AAS (Angle-Angle-Side) Theorem: - If two angles and a non-included side of one triangle are congruent to the corresponding two angles and non-included side of another triangle, the triangles are congruent.
HL (Hypotenuse-Leg) Theorem: - Applicable only to right-angled triangles. If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, the triangles are congruent.

Definition of Congruent Shapes: - Two shapes are congruent if one can be transformed into the other through a sequence of rigid motions (isometries), which include translations, reflections, and rotations. - Congruent shapes have the exact same dimensions and interior angles. Corresponding segments have equal lengths, and corresponding angles have equal measures.
CPCTC (Corresponding Parts of Congruent Triangles are Congruent): - This acronym is a fundamental logic tool used in formal proofs after triangle congruence has been established. - Once two triangles are proven congruent using criteria such as SSS, SAS, ASA, AAS, or HL, it follows logically that every other corresponding pair of parts (sides or angles) not used in the initial proof is also congruent. - For example, if $\Delta ABC \cong \Delta DEF$ by SAS, then it can be concluded by CPCTC that the third side (side $AC$ ) is congruent to its corresponding side (side $DF$ ).