Triangle Tactics Slides Advanced Geometry & Trig
Unit: Non-Right Trigonometry
Triangle Tactics:
Law of Sines & Cosines
Solving oblique triangles when standard right-triangle trigonometry no longer applies.
Classify
AAS, ASA, SAS, SSS
Decide
Select Sines or Cosines
Apply
Model Real Navigation
The Oblique Dilemma
FOUNDATIONAL CONCEPT
Right Triangles
One \(90^\circ\) Angle
Standard tools work:
• Pythagorean: \(a^2 + b^2 = c^2\)
• \(\sin \theta = \frac{\text{opp}}{\text{hyp}}\) , \(\cos \theta = \frac{\text{adj}}{\text{hyp}}\)
• Clear hypotenuse opposite \(90^\circ\)
Fast and direct, but strictly requires a right angle.
Oblique Triangles
No \(90^\circ\) Angle
SOH CAH TOA Fails!
No hypotenuse exists. We need universal laws that work on any triangle on earth.
Solution: Law of Sines & Law of Cosines
Standard notation: Angles \(A, B, C\) sit opposite sides \(a, b, c\).
Sum: \(A + B + C = 180^\circ\)
Law of Sines: Proportional Pairing
Ratio Pairing
The ratio of side length to opposite angle sine is constant:
\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]
Requires at least ONE complete opposite pair: (Angle + Side).
Case 1: AAS
Two angles and one non-included side given.
Case 2: ASA
Two angles and included side given. Find 3rd angle first!
Tactical Rule: When you know two angles, you know all three: \(C = 180^\circ - (A + B)\). Always Pair Up
Model 1: Solving with Law of Sines
Case: AAS
Given Triangle
\(A = 35^\circ\)
\(B = 75^\circ\)
\(a = 12\text{ cm}\)
Goal: Find angle \(C\) and side \(b\).
1
Find Missing Angle \(C\)
\(C = 180^\circ - (35^\circ + 75^\circ) = 70^\circ\)
2
Law of Sines Ratio for Side \(b\)
\(\frac{b}{\sin(75^\circ)} = \frac{12}{\sin(35^\circ)} \implies b = \frac{12 \cdot \sin(75^\circ)}{\sin(35^\circ)}\)
3
Calculate & Reasonableness Check
\(b \approx \frac{12 \cdot 0.9659}{0.5736} \approx 20.21\text{ cm}\) \(b > a \ \checkmark\)
Larger angle is always opposite the longer side (\(75^\circ > 35^\circ \implies b > a\)). Solved: \(C = 70^\circ, b \approx 20.2\text{ cm}\)
Checkpoint 1: Law of Sines Quick-Check
Fill in Guided Notes
Triangle Scenario
In \(\triangle DEF\):
• Angle \(D = 44^\circ\)
• Angle \(E = 62^\circ\)
• Side \(d = 18\text{ ft}\)
Look at your Guided Notes page 1: Complete questions 1, 2, and 3 now!
Q1
What is angle \(F\)?
\(F = 180^\circ - (44^\circ + 62^\circ) = 74^\circ\)
Q2
Which side is longer: \(d\) or \(e\)?
Side \(e > d\) because opposite angle \(E (62^\circ) > D (44^\circ)\).
Q3
What is the exact ratio to solve for side \(e\)?
\(\frac{e}{\sin(62^\circ)} = \frac{18}{\sin(44^\circ)} \implies e \approx 22.9\text{ ft}\)
Sanity Check: Side \(e\) (\(22.9\text{ ft}\)) is indeed longer than side \(d\) (\(18\text{ ft}\)). Checkpoint Passed ✓
Law of Cosines: Generalized Pythagorean
No Opposite Pairs
When no opposite pair is known, the Law of Cosines breaks the logjam:
\[c^2 = a^2 + b^2 - 2ab \cos C\]
If \(C = 90^\circ\), \(\cos(90^\circ) = 0 \implies c^2 = a^2 + b^2\)!
Finding Angle: \(\cos C = \frac{a^2 + b^2 - c^2}{2ab}\)
Case 1: SAS
Two sides and included angle known. Compute 3rd side first.
Case 2: SSS
All three sides known. Find largest angle first!
PEMDAS Caution: In \(a^2 + b^2 - 2ab \cos C\), multiply \(2ab\) by \(\cos C\) before subtracting! PEMDAS
Model 2: Solving with Law of Cosines
Case: SAS
Given Triangle
\(a = 8\text{ m}\)
\(b = 11\text{ m}\)
\(C = 52^\circ\)
Notice: Angle \(C\) is sandwiched between \(a\) and \(b\). No opposite pair!
1
Substitute into Formula
\(c^2 = 8^2 + 11^2 - 2(8)(11)\cos(52^\circ)\)
2
Evaluate Following Order of Operations
\(c^2 = 64 + 121 - 176(0.6157) = 185 - 108.36 = 76.64\)
3
Take Square Root
\(c = \sqrt{76.64} \approx 8.75\text{ m}\) Now switch to Sines!
Tactical Strategy: With side \(c\) found, pair \((c, C)\) is complete. Side \(c \approx 8.75\text{ m}\)
Checkpoint 2: Law of Cosines Trap-Check
Fill in Guided Notes
Spot the Blunder
A Student's Work:
\(c^2 = 5^2 + 8^2 - 2(5)(8)\cos(50^\circ)\)
\(c^2 = 89 - 80\cos(50^\circ)\)
\(c^2 = 9 \cdot \cos(50^\circ) \approx 5.79\) ✗
Question: What illegal mathematical operation did the student perform? Explain in your notes!
SSS Rule Test
Triangle with Sides:
\(a = 12\text{ cm}, b = 17\text{ cm}, c = 23\text{ cm}\)
1. Which angle must you solve for first?
→ Angle \(C\) (opposite longest side \(c=23\)).
2. Why is this critical?
If the triangle contains an obtuse angle, it will be \(\angle C\). \(\cos^{-1}\) can produce obtuse angles (\(>90^\circ\)), but \(\sin^{-1}\) cannot!
Always preserve multiplication precedence: \(89 - (80 \cdot \cos 50^\circ)\). Trap Avoided ✓
The Tactical Decision Matrix
WHICH LAW DO I CHOOSE?
Law of Sines Pair Known
AAS (Angle-Angle-Side) Law of Sines
ASA (Angle-Side-Angle) Law of Sines
Key Clue: 2 known angles means you have all 3 angles! Set up direct ratio.
Law of Cosines No Pairs
SAS (Side-Angle-Side) Law of Cosines
SSS (Side-Side-Side) Law of Cosines
Key Clue: Not enough angles for a pair. Use Cosines first, then switch to Sines.
Quick Test: \(A = 42^\circ, B = 68^\circ, c = 15\) → ASA (Sines) Quick Test: \(a = 7, b = 9, c = 12\) → SSS (Cosines)
Real-World Scenario: Wildfire Triangulation
Surveying Model
Field Scenario
Two ranger towers, Alpha and Bravo, are 10 miles apart along a baseline. Both spot a wildfire at point \(F\).
• \(\angle A = 54^\circ\), \(\angle B = 68^\circ\)
• Baseline side \(c = 10\text{ miles}\)
Find distance from Tower Alpha to the Fire (side \(b\)).
Classification: ASA → Law of Sines
\(\angle F = 180^\circ - (54^\circ + 68^\circ) = 58^\circ\)
Set up ratio for side \(b\) (opposite \(\angle B\)):
\(\frac{b}{\sin(68^\circ)} = \frac{10}{\sin(58^\circ)} \implies b = \frac{10 \cdot \sin(68^\circ)}{\sin(58^\circ)}\)
\(b \approx \frac{10 \cdot 0.9272}{0.8480} \approx 10.93\text{ miles}\)
Result: The wildfire is approx 10.9 miles from Tower Alpha.
Triangulation powers search-and-rescue, GPS, and surveying navigation. Distance \(\approx 10.9\text{ mi}\)
Tactical Debrief: Check Your Radar
ACTIVE PRACTICE
Scenario A
Bridge Truss
\(a = 14\text{ m}, b = 19\text{ m}, c = 22\text{ m}\)
Which Law First?
Law of Cosines (SSS)
Find largest \(\angle C\) first!
Scenario B
Ship Course
\(A = 48^\circ, B = 62^\circ, a = 35\text{ nm}\)
Which Law First?
Law of Sines (AAS)
Pair \((a, A)\) is ready.
Scenario C
Flight Radar
\(b = 280\text{ km}, c = 340\text{ km}, A = 40^\circ\)
Which Law First?
Law of Cosines (SAS)
Compute side \(a\) first.
Guided Notes complete! Ready for independent tactical practice on the worksheet. Begin Worksheet
Triangle Tactics Worksheet Trigonometry & Spatial Geometry
Triangle Tactics Worksheet
Navigating Oblique Triangles with the Law of Sines & Law of Cosines
Name:
Date:
Period:
Score:
Law of Sines: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)
Law of Cosines: \(c^2 = a^2 + b^2 - 2ab\cos C\)
PART 1
The Strategic Radar: Classify & Select
Examine the given triangle parameters. State the classification (AAS, ASA, SAS, or SSS ) and identify which law must be used first .
#
Given Triangle Info
Classification
First Law to Apply
A
\(A = 40^\circ, B = 65^\circ, a = 14\text{ cm}\)
B
\(a = 9\text{ m}, b = 15\text{ m}, c = 18\text{ m}\)
C
\(b = 12\text{ ft}, c = 16\text{ ft}, A = 48^\circ\)
D
\(A = 51^\circ, B = 73^\circ, c = 20\text{ mm}\)
PART 2
Oblique Triangle Calculations
Show all complete algebraic steps. Round lengths to the nearest tenth and angles to the nearest whole degree.
Problem 1: In \(\triangle ABC\), \(A = 32^\circ\), \(B = 81^\circ\), and side \(a = 9.4\text{ cm}\).
Find: Side \(b\) and Angle \(C\)
A (32°) B (81°) C a = 9.4 b = ? Figure not drawn to scale
Angle \(C = \) Side \(b = \)
Problem 2: In \(\triangle PQR\), \(p = 7.0\text{ m}\), \(q = 11.0\text{ m}\), and included angle \(R = 64^\circ\).
Find: Side \(r\)
R (64°) P Q q = 11.0 p = 7.0 r = ? Figure not drawn to scale
Side \(r = \)
Triangle Tactics • High School Trigonometry Page 1 of 2
Triangle Tactics • Advanced Applications
Student Name:
Problem 3: Finding Angles with SSS
A triangle has side lengths \(a = 6\text{ cm}\), \(b = 9\text{ cm}\), and \(c = 13\text{ cm}\). Determine the measure of the largest angle .
Law of Cosines
Strategic Hint:
The largest angle always sits opposite the longest side. Identify which angle to solve for first!
Isolate \(\cos(\theta)\) before using \(\cos^{-1}\).
Longest side: \(c = 13\text{ cm}\) Largest Angle (\(\angle C\)) =
PART 3
Tactical Real-World Modeling
Read each scenario, sketch or label the triangle, choose the correct law, and solve with appropriate units.
Problem 4: Coast Guard Rescue Triangulation
Navigational Trig
Two coastal radar stations, Station Echo and Station Foxtrot, are located 15.0 nautical miles apart along a straight shoreline running East-West. Both stations track an offshore vessel in distress at point \(V\). The bearing from Station Echo to the vessel creates an angle of \(58^\circ\) with the shoreline. The bearing from Station Foxtrot to the vessel creates an angle of \(74^\circ\) with the shoreline.
1. Sketch & Setup:
(Sketch \(\triangle EFV\))
Determine \(\angle V\) first.
Dist. from Station Foxtrot to Vessel =
Problem 5: Civil Engineering - Ravine Tunnel Length
Surveying Model
Surveyors need to determine the distance \(d\) between two cliff markers \(A\) and \(B\) separated by an impassable canyon. From an accessible benchmark point \(C\), a transit instrument measures the distance to marker \(A\) as 240 meters and to marker \(B\) as 310 meters . The angle between the two lines of sight (\(\angle ACB\)) is measured to be 46.0^\circ .
Identify setup:
Sides: \(240\text{ m}, 310\text{ m}\)
Case: SAS
Tunnel distance \(d = \)
Triangle Tactics • High School Trigonometry Page 2 of 2
Triangle Tactics Answer Key Teacher Resource • Solutions & Rubric
Triangle Tactics Answer Key
Complete Step-by-Step Solutions & Grading Checkpoints
Answer Key Total Points: 25 pts
PART 1 KEY
The Strategic Radar: Classify & Select
(4 pts • 1 pt each)
#
Given Triangle Info
Classification
First Law to Apply
Rationale
A
\(A=40^\circ, B=65^\circ, a=14\text{ cm}\)
AAS
Law of Sines
Known pair \((a, A)\) exists immediately.
B
\(a=9\text{ m}, b=15\text{ m}, c=18\text{ m}\)
SSS
Law of Cosines
Zero angles known; no opposite pairs.
C
\(b=12\text{ ft}, c=16\text{ ft}, A=48^\circ\)
SAS
Law of Cosines
Angle \(A\) is included between \(b\) and \(c\).
D
\(A=51^\circ, B=73^\circ, c=20\text{ mm}\)
ASA
Law of Sines
Find \(C = 180^\circ - (51+73) = 56^\circ\) then pair \((c, C)\).
PART 2 KEY
Oblique Triangle Calculations
(10 pts • 5 pts each)
Problem 1 Solution: Law of Sines (\(A=32^\circ, B=81^\circ, a=9.4\text{ cm}\)) 5 Points
Step 1: Determine Angle \(C\) (Triangle Sum Theorem)
\(C = 180^\circ - (32^\circ + 81^\circ) = 180^\circ - 113^\circ = \mathbf{67^\circ}\)
[1 pt]
Step 2: Set Up Law of Sines for Side \(b\)
\(\frac{b}{\sin(81^\circ)} = \frac{9.4}{\sin(32^\circ)} \implies b = \frac{9.4 \cdot \sin(81^\circ)}{\sin(32^\circ)}\)
\(b \approx \frac{9.4 \cdot 0.98769}{0.52992} \approx \frac{9.2843}{0.52992} \approx \mathbf{17.5\text{ cm}}\)
[3 pts setup/calc]
Self-Check: Largest angle \(81^\circ\) opposes largest side \(17.5\text{ cm}\). Answers: \(C = 67^\circ\), \(b \approx 17.5\text{ cm}\)
Problem 2 Solution: Law of Cosines SAS (\(p=7.0\text{ m}, q=11.0\text{ m}, R=64^\circ\)) 5 Points
Step 1: Formula Setup for Missing Opposite Side \(r\)
\(r^2 = p^2 + q^2 - 2pq\cos R = 7.0^2 + 11.0^2 - 2(7.0)(11.0)\cos(64^\circ)\)
[1 pt]
Step 2: Arithmetic Evaluation (PEMDAS Check)
\(r^2 = 49 + 121 - 154 \cdot \cos(64^\circ) = 170 - 154(0.43837)\)
\(r^2 = 170 - 67.509 = 102.491\)
[2 pts]
Step 3: Solve for \(r\)
\(r = \sqrt{102.491} \approx \mathbf{10.1\text{ m}}\)
[2 pts]
Common Error: Do NOT calculate \((170 - 154)\cos 64^\circ = 16\cos 64^\circ\)! Answer: \(r \approx 10.1\text{ m}\)
Triangle Tactics Answer Key • Page 1 of 2 Page 1 of 2
Triangle Tactics Answer Key • Part 3 & Modeling
Remaining: 11 Points
Problem 3 Solution: SSS Finding Largest Angle (\(a=6, b=9, c=13\)) 3 Points
Step 1: Identify Target Angle [1 pt]
The longest side is \(c = 13\text{ cm}\), so the largest angle is angle \(C\).
Step 2: Law of Cosines Rearrangement [1 pt]
\(c^2 = a^2 + b^2 - 2ab\cos C \implies \cos C = \frac{a^2 + b^2 - c^2}{2ab}\)
\(\cos C = \frac{6^2 + 9^2 - 13^2}{2(6)(9)} = \frac{36 + 81 - 169}{108} = \frac{117 - 169}{108} = \frac{-52}{108} \approx -0.4815\)
Step 3: Solve Inverse Cosine
\(C = \cos^{-1}(-0.4815) \approx 118.78^\circ \approx \mathbf{119^\circ}\) (obtuse!)
Answer: \(C \approx 119^\circ\)
Problem 4 Solution: Coast Guard Triangulation (ASA / Sines) 4 Points
Step 1: Triangle Angles Calculation [1 pt]
In \(\triangle EFV\), \(\angle E = 58^\circ\), \(\angle F = 74^\circ\).
Step 2: Select Side & Law of Sines Setup [2 pts]
Distance from Station Foxtrot to Vessel is side \(e\) (opposite \(\angle E\)):
\(\frac{e}{\sin(58^\circ)} = \frac{15.0}{\sin(48^\circ)} \implies e = \frac{15.0 \cdot \sin(58^\circ)}{\sin(48^\circ)}\)
\(e \approx \frac{15.0 \cdot 0.84805}{0.74314} \approx \frac{12.7207}{0.74314} \approx \mathbf{17.1\text{ nautical miles}}\)
Note: If student solved for Station Echo distance \(f\): \(f = \frac{15\sin 74^\circ}{\sin 48^\circ} \approx 19.4\text{ nm}\). Answer: \(\mathbf{17.1\text{ nm}}\)
Problem 5 Solution: Canyon Tunnel Length (SAS / Cosines) 4 Points
Step 1: Geometric Setup & Formula [1 pt]
In \(\triangle ABC\), \(a = 310\text{ m}\), \(b = 240\text{ m}\), included angle \(\angle C = 46.0^\circ\). Missing canyon distance is \(d = c\).
Step 2: Substitution & Evaluation [2 pts]
\(d^2 = 240^2 + 310^2 - 2(240)(310)\cos(46^\circ)\)
\(d^2 = 57,600 + 96,100 - 148,800(0.69466) = 153,700 - 103,365.1 = 50,334.9\)
\(d = \sqrt{50,334.9} \approx \mathbf{224.4\text{ meters}}\)
Acceptable rounding range: \(224.2\text{ m}\) to \(224.5\text{ m}\) based on intermediate precision. Answer: \(\mathbf{224.4\text{ meters}}\)
Triangle Tactics Answer Key • High School Trigonometry Page 2 of 2
Triangle Tactics Teacher Guide Teacher Facilitation Guide
Triangle Tactics: Teacher Guide
Navigating Oblique Triangles with Law of Sines & Law of Cosines
High School Trig / Pre-Calc
Pacing: 60–90 Mins
Standards & Materials
CCSS.MATH HSG.SRT.D.10 & D.11
• Slides: 11-Slide Interactive Presentation
• Notes: 2-Page Student Guided Notes
• Practice: 2-Page Worksheet + Key
• Assessment: 2-Up Exit Ticket (Form A/B)
Core Learning Objectives
• Classify: Identify triangle cases (AAS, ASA, SAS, SSS) from given parameters.
• Decide: Determine whether Law of Sines or Cosines is the mandatory first step.
• Solve: Calculate missing sides/angles with precision and algebraic rigor.
• Model: Formulate and solve real navigational and surveying scenarios.
STRUCTURE
Lesson Flow & Pacing Architecture
Pacing
Phase
Slide & Handout Match
Key Teacher Actions
0–10m
Hook & The Dilemma
Slides 1–2 • Notes Section 1
Distribute Guided Notes. Ask why SOH CAH TOA fails on non-right triangles; record definition.
10–25m
Sines & Checkpoint 1
Slides 3–5 • Notes Sections 2–4
Model AAS on Slide 4. Pause at Slide 5 Checkpoint 1 for students to solve in Guided Notes before revealing.
25–45m
Cosines & Checkpoint 2
Slides 6–8 • Notes Sections 5–7
Teach Cosines formula. Pause at Slide 8 Checkpoint 2 to address the PEMDAS trap and SSS angle order.
45–55m
Decider & Modeling
Slides 9–10 • Notes Section 8
Review Decision Matrix; connect to wildfire triangulation scenario.
55–78m
Worksheet Practice
Student Worksheet • Parts 1–3
Students transition to independent/paired worksheet; circulate with Answer Key.
78–90m
Exit Ticket & Closure
Exit Ticket (Form A/B) + Slide 11
Administer 2-up exit ticket (5–8 min). Collect for rapid formative assessment.
DIALOGUE
High-Leverage Questioning Stems
To Prompt Law Selection:
• "Do you have an angle and the exact side opposite to it?"
• "If you know two angles, what do you automatically know about the third?"
• "Why is it impossible to begin SSS with the Law of Sines?"
To Enforce Reasonableness:
• "Which angle is largest? Is your calculated opposite side also the longest?"
• "Could an oblique triangle have two angles greater than \(90^\circ\)?"
• "Why does \(\cos(C)\) becoming negative indicate an obtuse angle?"
Triangle Tactics • Instructional Facilitation Guide Page 1 of 2
Diagnostic Interventions & Exit Ticket Key
Triangle Tactics • Page 2
DIAGNOSTICS
Common Student Misconceptions & Corrections
1. PEMDAS Violation
Calculating \((a^2+b^2-2ab)\cos C\).
Fix: Box \([2ab\cos C]\) as single multiplication block.
2. Radian Mode Error
Evaluating degrees in radian mode.
Fix: Opening check: verify \(\sin(30^\circ) = 0.5\).
3. Missing Pair in ASA
Assuming Sines cannot be used.
Fix: Find 3rd angle first via \(180^\circ - (A+B)\).
EXIT KEY
Exit Ticket Quick Grading Key (Forms A & B)
10 Pts Total
Form A Solutions \(a=14, b=19, C=58^\circ\)
1. Classify: SAS • First Law: Law of Cosines (3 pts)
2. Calc: \(c^2 = 14^2 + 19^2 - 2(14)(19)\cos 58^\circ = 196 + 361 - 532(0.5299) = 557 - 281.9 = 275.1 \implies \mathbf{c \approx 16.6\text{ m}}\) (5 pts)
3. Error: \(110^\circ + 80^\circ = 190^\circ > 180^\circ\). Sum exceeds triangle limit (2 pts).
Form B Solutions \(P=41^\circ, Q=73^\circ, p=12\text{ cm}\)
1. Classify: AAS • First Law: Law of Sines (3 pts)
2. Calc: \(\frac{q}{\sin 73^\circ} = \frac{12}{\sin 41^\circ} \implies q = \frac{12 \cdot 0.9563}{0.6561} = \frac{11.4756}{0.6561} \implies \mathbf{q \approx 17.5\text{ cm}}\) (5 pts)
3. Error: Sines requires at least 1 angle-side pair. SSS has 0 angles; must start with Cosines (2 pts).
RUBRIC
4-Point Mastery Criteria
Level
Classification & Setup
Algebraic Execution
Context & Precision
4 - Mastery
Identifies AAS/SAS/SSS accurately; chooses optimal law every time.
Clean algebraic isolation; zero order-of-operations errors in Cosines.
Appropriate units; correct rounding to tenths/degrees.
3 - Proficient
Selects correct law; labels opposite pairs correctly.
Minor computation slips, but correct algebraic formula structure.
Missing units or minor rounding error.
2 - Developing
Struggles to distinguish SAS from SSS or AAS from ASA.
Commits PEMDAS error in Cosines or inverts sine ratios.
Work incomplete or lacks geometric diagram.
Triangle Tactics • High School Trigonometry Page 2 of 2
Triangle Tactics Guided Notes Slide Presentation Companion
Triangle Tactics Guided Notes
Active Listening & Practice Companion: Law of Sines & Law of Cosines
Name:
Date:
Period:
Teacher:
SLIDE 2
1. The Oblique Dilemma
Right Triangles:
Contains exactly one angle.
Can use: Pythagorean Theorem & (SOH CAH TOA).
Oblique Triangles:
Contains NO angle.
Why does SOH CAH TOA fail? There is no !
Standard Convention: Angle \(A\) opposes side , \(B\) opposes , \(C\) opposes . Angle Sum: \(A + B + C = \)
SLIDE 3
2. Law of Sines: Proportional Ratio
Write the Complete Law of Sines Formula:
(Write the three equal ratios here)
Core Requirement: To use this law, you must know at least one complete .
Applicable Cases:
• AAS: Angle-Angle-Side (Non-included)
• ASA: Angle-Side-Angle (Included side)
Golden Rule: Knowing two angles means you know the 3rd via \(180^\circ - (A+B)\)!
SLIDE 4
3. Slide Model 1 Walkthrough (AAS)
Given: \(A = 35^\circ\), \(B = 75^\circ\), \(a = 12\text{ cm}\) Find: Angle \(C\) and Side \(b\)
Step 1: Calculate Angle \(C\)
Angle \(C = \) _________
Step 2: Proportion for Side \(b\)
Side \(b \approx \) _________
SLIDE 5
4. Tactical Checkpoint 1: Active Try-It
Check for Understanding
In \(\triangle DEF\): \(D = 44^\circ\), \(E = 62^\circ\), side \(d = 18\text{ ft}\) Case: AAS
Q1: Angle \(F\) measure
\(F = \) __________
Q2: Which side is longer?
Reason: __________
Q3: Ratio to solve \(e\)
\(e \approx \) __________
Triangle Tactics Guided Notes • High School Trigonometry Page 1 of 2
Triangle Tactics Guided Notes • Law of Cosines & Synthesis
Student Name:
SLIDE 6
5. Law of Cosines: Generalized Pythagorean
Primary Formula (Solving for Side \(c\)):
\(c^2 = \) ___________________________
Rearranged for Angle \(C\):
\(\cos C = \) ___________________________
When to Deploy Cosines:
• SAS: Two sides and included angle.
• SSS: All three sides known, zero angles.
Notice: If \(C = 90^\circ\), \(\cos(90^\circ) = 0\), turning into the standard Pythagorean theorem!
SLIDE 7
6. Slide Model 2 Walkthrough (SAS)
Given: \(a = 8\text{ m}\), \(b = 11\text{ m}\), \(C = 52^\circ\) Goal: Find side \(c\)
Step 1: Formula Setup & Substitution
Step 2: PEMDAS Multiplication & Square Root
Side \(c \approx \) _________
SLIDE 8
7. Tactical Checkpoint 2: Trap Analysis & SSS
Check for Understanding
A. Spot the Blunder:
\(89 - 80\cos(50^\circ) \rightarrow 9\cos(50^\circ)\) ✗
Why is this subtraction strictly illegal?
B. SSS Strategy Test:
Given triangle with sides \(12, 17, 23\):
Which angle must you find first, and why?
SLIDE 9
8. The Tactical Decision Matrix (Summary)
Triangle Case
Opposite Pair Known?
First Law to Use
Key First Step
AAS
YES
Law of Sines
Find 3rd angle via \(180^\circ - (A+B)\)
ASA
YES (after 3rd angle)
Law of Sines
Find opposite angle first to complete pair
SAS
NO
Law of Cosines
Find missing opposite side length
SSS
NO
Law of Cosines
Find LARGEST angle first (opposite longest side)
Slide 10 Triangulation Takeaway: Knowing a baseline distance + two viewing angles (ASA) allows full coordinate location!
Triangle Tactics Guided Notes • High School Trigonometry Page 2 of 2
Triangle Tactics Exit Ticket FORM A
Triangle Tactics: Exit Ticket
Name: Period: / 10 pts
1. Triangle Classification & Law Selection (3 pts):
In \(\triangle ABC\), you are given: \(a = 14\text{ m}, b = 19\text{ m}, C = 58^\circ\) .
Classification (AAS, ASA, SAS, SSS):
First Law to Apply:
2. Tactical Calculation (5 pts):
Using the given info above, calculate the length of side \(c\) to the nearest tenth.
Show setup & calculation:
Side \(c \approx \)
3. Error Analysis (2 pts):
A classmate says: "In \(\triangle ABC\), \(A=110^\circ\) and \(B=80^\circ\)." Why is this mathematically impossible?
Confidence Level: ○ 1 (Lost) • ○ 2 (Shaky) • ○ 3 (Solid) • ○ 4 (Mastered) Form A
Fold or cut along dashed line
FORM B
Triangle Tactics: Exit Ticket
Name: Period: / 10 pts
1. Triangle Classification & Law Selection (3 pts):
In \(\triangle PQR\), you are given: \(P = 41^\circ, Q = 73^\circ, p = 12\text{ cm}\) .
Classification (AAS, ASA, SAS, SSS):
First Law to Apply:
2. Tactical Calculation (5 pts):
Using the given info above, calculate the length of side \(q\) to the nearest tenth.
Show setup & calculation:
Side \(q \approx \)
3. Error Analysis (2 pts):
A classmate says: "I can always use Law of Sines if I know three side lengths (SSS)." Why is this incorrect?
Confidence Level: ○ 1 (Lost) • ○ 2 (Shaky) • ○ 3 (Solid) • ○ 4 (Mastered) Form B
