Law of Sines (AAS Case)

Flashcards covering the foundational concepts of the Law of Sines, the distinction between oblique and right triangles, and solving for parts of a triangle using the AAS case.

What are the two types of triangles that fall under the category of oblique triangles?

Acute triangle and obtuse triangle.

Why do basic trigonometric ratios (SOH-CAH-TOA) not work for oblique triangles?

Because basic trigonometric ratios only apply to right triangles.

Which three cases of triangles can be solved using the Law of Sines?

Case 1: AAS (Angle-Angle-Side), Case 2: ASA (Angle-Side-Angle), and Case 3: SSA (Side-Side-Angle).

In a right triangle, what is the formula for the six trigonometric ratios based on SOH-CAH-TOA?

$\sin(\theta) = \frac{\text{opp}}{\text{hyp}}$, $\cos(\theta) = \frac{\text{adj}}{\text{hyp}}$, $\tan(\theta) = \frac{\text{opp}}{\text{adj}}$, $\csc(\theta) = \frac{\text{hyp}}{\text{opp}}$, $\sec(\theta) = \frac{\text{hyp}}{\text{adj}}$, and $\cot(\theta) = \frac{\text{adj}}{\text{opp}}$.

What defines the AAS (Angle-Angle-Side) condition?

The condition requires knowing the measurements of two angles and a side that is not included between those two angles.

What is meant by an "included side" in a triangle?

It is a side that lies between two specific angles.

What is meant by an "included angle" in a triangle?

It is an angle formed between two specific sides.

If the goal is to find a missing side in triangle $ABC$, which form of the Law of Sines is used?

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

If the goal is to find a missing angle in triangle $ABC$, which form of the Law of Sines is used?

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

In the derivation of the Law of Sines, if a perpendicular $h$ is dropped from vertex $A$ to side $a$, what are the values of $\sin(B)$ and $\sin(C)$?

$\sin(B) = \frac{h}{c}$ and $\sin(C) = \frac{h}{b}$

Based on the derivation using altitude $h$, what equation relates sides $b$ and $c$ with their opposite angles?

$$c \times \sin(B) = b \times \sin(C)$$

In a triangle with $A = 92^{\circ}$, $B = 28^{\circ}$, and side $a = 15\,yd$, what is the calculation for side $b$ ($AC$)?

$$AC = \frac{15(\sin(28^{\circ}))}{\sin(92^{\circ})}$$

Using the Law of Sines for the AAS case where $A = 59^{\circ}$, $C = 15^{\circ}$, and $c = 10\,yd$, what is the setup to find side $a$ ($BC$)?

$$\frac{BC}{\sin(59^{\circ})} = \frac{10}{\sin(15^{\circ})}$$

What is the approximate resulting value for side $BC$ given $c = 10\,yd$, $A = 59^{\circ}$, and $C = 15^{\circ}$?

$$33.1233\,yd$$