Exhaustive Study Guide: The Law of Sines and Triangular Proportions

Homework Review and Challenge Results

• Homework Clarification: The session began with a check for homework questions.     * One student inquired about a specific problem requiring a radius or degree setting on the calculator.     * Calculation detail for a problem: $\frac{1}{2} \times 50 \times 24 \times \sin(150^{\circ})$.

• Sarah and Matthew Challenge Results:     * The percentage for Sarah was calculated as $34.8 \%$.     * Other results mentioned for different parts of the challenge included $16 \%$, $28.3 \%$, and $36.9 \%$.

Derivation of the Law of Sines

• Recap of Triangle Area Formulas: The area of a triangle, denoted as $K$, can be expressed in three ways depending on the known sides and the included angle (SAS):     * $K = \frac{1}{2}ab \sin(C)$     * $K = \frac{1}{2}bc \sin(A)$     * $K = \frac{1}{2}ac \sin(B)$

• Connecting the Formulas: Since all three expressions represent the area of the exact same triangle, they are equal to each other:     * $\frac{1}{2}bc \sin(A) = \frac{1}{2}ac \sin(B) = \frac{1}{2}ab \sin(C)$

• Simplification Steps:     1. Eliminate the fractions by multiplying the entire equation by $2$:         $bc \sin(A) = ac \sin(B) = ab \sin(C)$     2. To isolate $\sin(A)$, divide the expressions. For example, dividing by $bc$ results in:         $\sin(A) = \frac{ac \sin(B)}{bc} = \frac{a \sin(B)}{b}$     3. By rearranging these relationships using the "keep-change-flip" method for division, we arrive at the standard format for the Law of Sines.

• The Law of Sines:     * $\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}$     * Alternatively, the reciprocal version is equally valid:         $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$

• Core Principle: The sides and angles of a triangle are always proportional.     * The side opposite the largest angle is always the largest side.     * The side opposite the smallest angle is always the smallest side.

Law of Sines vs. Law of Cosines

• Law of Cosines Usage: Required when dealing with three sides and one angle in play (where one of the four is the unknown).

• Law of Sines Usage: Required when dealing with two sides and two angles in play (where one of the four is the unknown).

Example Problems and Scenarios

• Example 1: Triangle ABC     * Given: side $a = 8$, $\angle A = 30^{\circ}$, $\angle C = 55^{\circ}$.     * Objective: Find side $c$ to the nearest tenth.     * Equation: $\frac{8}{\sin(30^{\circ})} = \frac{c}{\sin(55^{\circ})}$     * Procedure: Cross-multiply using the "butterfly method": $c = \frac{8 \times \sin(55^{\circ})}{\sin(30^{\circ})}$.     * Note: $\sin(30^{\circ}) = \frac{1}{2}$.     * Result: $c \approx 13.1\,\text{units}$.     * Sanity Check: Since 55^{\circ} > 30^{\circ}, the side $c$ (13.1) should be larger than side $a$ (8), which it is.

• Example 2: Triangle DEF     * Given: $\sin(F) = \frac{1}{5}$, $\sin(D) = \frac{2}{5}$, and side $f = 24$.     * Objective: Find the length of side $d$.     * Equation: $\frac{d}{\sin(D)} = \frac{f}{\sin(F)}$     * Substitution: $\frac{d}{2/5} = \frac{24}{1/5}$     * Note: You do not take the sine of the fraction; the fraction itself is the value of the sine.     * Calculation: $d \times (1/5) = 24 \times (2/5) \rightarrow d/5 = 48/5$.     * Result: $d = 48$.

• Example 3: Ratio Problem (Triangle PQR)     * Given: $\sin(P) = \frac{1}{3}$ and $\sin(Q) = \frac{1}{4}$.     * Objective: Find the ratio of side $p$ to side $q$.     * Equation: $\frac{p}{\sin(P)} = \frac{q}{\sin(Q)} \rightarrow \frac{p}{1/3} = \frac{q}{1/4}$     * Procedure:         1. $3p = 4q$         2. Divide by $q$ to get the ratio: $\frac{p}{q} = \frac{4}{3}$.     * Result: The ratio is $4:3$.

• Example 4: Application (Triangle PQR)     * Given: $\angle PQR = 66.5^{\circ}$, $\angle QRP = 47^{\circ}$, side $RQ = 12.4\,\text{inches}$.     * Objective: Find side $PQ$ (labeled $x$) to the nearest tenth.     * Solving for Angle P: $\angle P = 180^{\circ} - (66.5^{\circ} + 47^{\circ}) = 66.5^{\circ}$.     * Equation: $\frac{x}{\sin(47^{\circ})} = \frac{12.4}{\sin(66.5^{\circ})}$     * Calculation Warning: Do not truncate or approximate values early in the calculator to avoid rounding errors.     * Result: $x \approx 9.9\,\text{inches}$.

• Example 5: Triangle CAT     * Given: side $c = 15.2$, side $a = 6.4$, $\angle C = 107.3^{\circ}$.     * Objective: Find $\angle A$ to the nearest tenth.     * Calculation Input: A student provided the raw calculation as $10.2203798998988$.     * Final Approximation: $\angle A \approx 23.7^{\circ}$.

Logistics and Upcoming Schedule

• Final Exam Preparation:     * There will be 10 review packets provided to cover all topics for the final.     * Each packet contains approximately 30 questions.     * These will be counted as homework assignments.     * The instructor intends to post them around the day of the test for completion over Memorial Day weekend.     * Note: Copy machines are currently malfunctioning and there are no staples, so packets may be posted online rather than handed out physically.

• Upcoming Curriculum:     * Thursday: The Ambiguous Case of the Law of Sines.     * Friday: Mixing Law of Sines and Law of Cosines.     * Monday: Forces with parallelograms.     * Tuesday: Review session.     * Wednesday: Unit Test.

• Assignments: Delta Math is due next Wednesday (originally posted as Monday, but corrected to the day of the test).

Questions & Discussion

• Student Question: "How did I get what? I don't know which one you're talking about."

• Teacher Response: Explained that the student simply needs to plug the values into the formula and checked if the calculator was in radians or degrees.

• Student Question: Regarding Example 4: "Would that make the bridge… stop from falling down?"

• Teacher Response: The teacher noted that you wouldn't want a bridge you are driving across to be off by even a tenth of an inch, noting that "should be" and "will be" are very different in engineering.

• Student Question: "Why are you trying to get… [the ratio]?"

• Teacher Response: The teacher explained the goal is to show every possible scenario of Law of Sines questions because the basic application is otherwise too easy. A ratio indicates how $p$ relates to $q$ rather than seeking a numerical value for the side length.

• Student Question: Inquiry about the naming of "Triangle CAT."

• Teacher Response: The teacher used a "weird" name like CAT to avoid students asking why she used specific initials, noting that triangle labels are arbitrary.