Laws of Sines and Cosines Lecture Review

Flashcards covering the definitions, formulas, and application cases for the Law of Sines and the Law of Cosines as presented in the lecture.

Law of Sines

A mathematical rule used to solve for missing sides or angles in a triangle when there is a known matching pair of a side and its opposite angle.

Law of Sines Formula

$$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$$

Side-Angle Correspondence

The principle that in the Law of Sines, the side and the angle that are opposite each other form a pair, identified by the same letter (Side $a$ to Angle $A$, Side $b$ to Angle $B$, Side $c$ to Angle $C$).

Case 1: Finding a Side (Law of Sines)

Using the Law of Sines when two angles and one side are given, such as finding $b$ when $A = 30^\circ$, $B = 45^\circ$, and $a = 8$, which results in $b \approx 11.31$.

Case 2: Finding an Angle (Law of Sines)

Using the Law of Sines when two sides and one angle are given, such as finding Angle $B$ when $a = 8$, $b = 11.31$, and $A = 30^\circ$, which results in $B \approx 45^\circ$.

Trigonometric values for Sine

In logic calculations, $\sin(30^\circ) = 0.5$ and $\sin(45^\circ) \approx 0.7071$.

Law of Cosines

A formula used for triangles that are not right triangles (those without a $90^\circ$ angle) to find a missing side or angle.

Law of Cosines Formula (Side)

$$c^2 = a^2 + b^2 - 2ab \cos(C)$$

Law of Cosines Formula (Angle)

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

Case 1: Find Missing Side (Law of Cosines)

Used when you have 2 sides and 1 included angle; for example, given Side $a = 5$, Side $b = 8$, and Angle $C = 60^\circ$, the calculation $c^2 = 25 + 64 - 40 = 49$ results in Side $c = 7$.

Case 2: Find Missing Angle (Law of Cosines)

Used when all 3 sides are known; for example, given $a = 5$, $b = 6$, and $c = 7$, the calculation yields $\cos(C) = 0.2$, which leads to Angle $C \approx 78.5^\circ$.

Angle Determination Step

The process of using a calculator to find an angle by pressing SHIFT or 2nd followed by the trigonometric function to get inverse values like $\sin^{-1}$ or $\cos^{-1}$.

Law of Cosines Strategy

Treated like a recipe where provided numbers replace the letters in the formula, followed by the order of operations to solve step-by-step.