Laws of Sines, Cosines & Triangle Tests – Comprehensive Study Notes

Fundamental proportionality for any (acute, obtuse, or right) triangle:
$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R$
• $a,\,b,\,c$ = side lengths opposite angles $A,\,B,\,C$
• $R$ = radius of the triangle’s circumscribed circle
Works even when a right angle is present; the right‐angle case is simply a subset of the general law.
Provides a direct route to solving AAS (angle–angle–side) or ASA (angle–side–angle) configurations.
When applied to right triangles, one of the three terms simplifies because $\sin(90^\circ)=1$.

"Click the blue triangle" prompts learners to toggle between acute, right, and obtuse renderings, reinforcing that the law’s ratios remain valid while the visible shape morphs.
Arrows provide a step-by-step rewind/advance of side–angle relationships, letting students observe that each ratio is constant despite side swapping.

Stated formula: $c^{2}=a^{2}+b^{2}-2ab\cos C$
Bridges the gap between right-triangle Pythagorean logic and oblique triangles.
• When $C=90^\circ,$ $\cos 90^\circ =0,$ and we recover $c^{2}=a^{2}+b^{2}.$
Practical classroom example ("swing & slide"):
• Two pieces of playground equipment are 100 ft apart, angle at the tree = $30^\circ$ .
• If the slide is 75 ft from the same tree, distance between swing & slide:
$d=\sqrt{100^{2}+75^{2}-2(100)(75)\cos(30^\circ)}\approx62.2\text{ ft}.$
• Shows why Cosine Law is the go-to when SAS (side–angle–side) data are known.

Transcript question: triangle with sides 3, 4, 6.
$3^{2}+4^{2}=25\neq6^{2}=36$
→ NOT a right triangle.
Possible determinations:
• Could be acute ($c^{2}

SSS, SAS, ASA, AAS
– Guarantee both equality of shape & size.
HL (Hypotenuse–Leg) for right triangles only: if hypotenuse & one leg match, triangles are congruent.
Every congruent‐triangle test automatically implies similarity, because congruence ⟹ similarity by definition.

AA (Angle–Angle): two equal angles are sufficient to declare similarity, but not congruence.
– The “additional” similarity test referred to in transcript; cannot ensure equal scale, only equal shape.
SAS Similarity: proportional side ratio + included angle equality.
SSS Similarity: three proportional side ratios.

Emphasizes that $\sin$ and $\cos$ reduce to opposite/hypotenuse and adjacent/hypotenuse only when one angle is $90^\circ$ .
Shows the neat cancellation in $\frac{c}{\sin 90^\circ}=c.$

Students asked to fill two aligned columns:
• Calculation – numeric/ algebraic steps using Sine or Cosine Law.
• Reason – verbal justification (e.g., "ASA → Law of Sines", "SAS → Law of Cosines").
Example values seen: 44, 25 → likely degree measures or side lengths to be solved.

Reminder: HL is essentially the Pythagorean Theorem expressed as a congruence shortcut.
The slide encourages linking the topic back to earlier units on similarity: "similar right triangles will be congruent if…"
• Emphasises that identical numeric hypotenuse & leg lengths collapse any scale factor $k$ to 1.

Safety design of playgrounds (windblown tree anchoring, swing & slide distances) demonstrates how trig ensures secure spacing.
Multiple representation strategy (interactive slide, arrows, color‐coded sides) aims at Universal Design for Learning (UDL)—supporting diverse visual & kinesthetic learners.

Page counters: "5 of 18", "91 432 Checks", etc. serve as micro-progress indicators; research shows chunked feedback improves retention.
Historical note "−3500 the dance" likely references earliest trigonometric tables emerging from ancient civilizations ≈3500 BCE.

Law of Sines: $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$
Law of Cosines: $c^{2}=a^{2}+b^{2}-2ab\cos C$
Pythagorean (right Δ): $a^{2}+b^{2}=c^{2}$
HL Congruence (right Δ): if $H{1}=H{2}\;\text{and}\;L{1}=L{2} \;\Rightarrow\; \triangle{1}\cong\triangle{2}.$