Trig Intervention Guide Trig Intervention Guide
Subject: High School Geometry | Topic: Laws of Sines & Cosines
Tier 2 Intervention
Learning Objective
Students will prove the Laws of Sines and Cosines using right triangle trigonometry and apply these laws to solve for missing sides and angles in oblique (non-right) triangles.
Standard: HS.G-SRT.D.10 - Prove the Laws of Sines and Cosines and use them to solve problems.
Common Misconceptions
• The "Right Angle" Trap: Attempting to use SOH CAH TOA directly on non-right triangles without creating an altitude.
• Law of Cosines Calculation Error: Incorrectly subtracting \(2ab \cos C\) before multiplying (order of operations).
• Case Confusion: Struggling to identify when to use Sines vs. Cosines based on given information (SAS vs. AAS).
Materials Needed
• Scientific Calculators
• Ruler/Straightedge
• Student "Sine & Cosine Lab" Worksheet
• "Surveying Success" Exit Ticket
Group Size
Ideal for 3-5 students to allow for frequent check-ins and shared derivation work.
Lesson Pacing & Facilitation
Phase Teacher Actions & Discussion Prompts 1. The Bridge (5 min) Prompt: "We know how to solve right triangles using SOH CAH TOA. But look at this triangle—no square box! How can we 'create' a right triangle inside it?"
Focus: Activating prior knowledge of altitudes and right triangle trig.
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| 2. The Proof (15 min) |
Guide students through the altitude derivation. Use the slide deck to show that \( h = b \sin A \) and \( h = a \sin B \). Set them equal.
Support: Have students highlight the two different right triangles in different colors on their lab sheet.
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| 3. Blueprint Strategy (15 min) |
Teach the "Law of Sines = Pairs" vs "Law of Cosines = Sandwich" mnemonic. Practice identifying the 'Law' before calculating.
Prompt: "Do we have a side and its opposite angle pair? If yes, use Sines!"
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| 4. Check for Success (5 min) |
Administer the "Surveying Success" Exit Ticket. Use the progress monitoring tool below to track mastery.
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Progress Monitoring Tracker
Use this table during the session to record student progress on key benchmarks.
Student Name Identify Altitudes Derive Law of Sines Select Correct Law Solve for Side Solve for Angle
Evidence of Mastery
• Correctly labels triangle parts (\(A, B, C\) vs \(a, b, c\)).
• Explains that Law of Cosines is like an "expanded Pythagorean Theorem."
• Successfully identifies SSS and SAS as Law of Cosines scenarios.
Intervention Notes Trig Blueprint Slides Trig Blueprints
Mastering the Laws of Sines & Cosines
The "Oblique" Problem
For right triangles, we have SOH CAH TOA and the Pythagorean Theorem.
But what if there is no right angle?
We need a new set of "blueprints" to find missing parts of oblique triangles.
A B C c b a
The Altitude Strategy
Derivation Step-by-Step
1 Drop an altitude h from vertex C.
2 In left triangle: \( \sin A = \frac{h}{b} \rightarrow h = b \sin A \)
3 In right triangle: \( \sin B = \frac{h}{a} \rightarrow h = a \sin B \)
4 Set them equal: \( b \sin A = a \sin B \)
A B C h b a
\[ \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} \]
The "Power Tool": Law of Cosines
The Blueprint
\( a^2 = b^2 + c^2 - 2bc \cos A \)
Looks like Pythagorean Theorem... with a twist!
When to use it?
SAS: The "Side-Angle-Side" Sandwich
SSS: You know all three sides
A Practical Tip
Think of the side \(a\) and angle \(A\) as the "Outer Frame."
They are at opposite ends of the formula. The other sides \(b\) and \(c\) do all the heavy lifting in the middle.
Strategic Blueprinting
LAW OF SINES
LOOK FOR
PAIRS
Angle + Opposite Side
Cases: AAS or ASA
LAW OF COSINES
LOOK FOR
SANDWICH
Sides + Included Angle
Cases: SAS or SSS
Trig Blueprint Worksheet Trigonometry Blueprint Lab
Solving Oblique Triangles Intervention
Name:
Date:
Part 1: The Foundation
Proving the Law of Sines
Fill in the blanks to complete the proof using the diagram to the right.
1. In the left right-triangle, using Sine:
2. Rearrange to solve for h:
3. In the right right-triangle, using Sine:
4. Rearrange to solve for h:
5. Since both expressions equal h, set them equal:
A B C h b a
Law of Sines: \( \frac{\sin A}{a} = \frac{\sin B}{b} \)
Part 2: The Decision Tree
Which Blueprint should I use?
Circle the correct law based on the given information. Do not solve yet!
Scenario 1
You know three sides (SSS)
Sines Cosines
Scenario 2
You have a "Side-Angle Pair"
Sines Cosines
Part 3: Solving the Specs
Problem 01: Sines
A=42° B=75° C c=22 a=?
Step 1: Set up the ratio
Step 2: Cross-multiply and solve for 'a'
Problem 02: Cosines
A=60° b=10 c=15 a=?
Step 1: Plug into the formula: \( a^2 = b^2 + c^2 - 2bc \cos A \)
Step 2: Calculate. Remember order of operations!
Surveying Success Exit Ticket Surveying Success
Exit Ticket: Oblique Triangles
Score
___ / 10
Student Name
Date
1. The Surveyor's Choice
A surveyor knows the lengths of two sides of a triangular field and the measure of the angle between them (Side-Angle-Side). Which law should they use to find the third side?
Law of Sines
Law of Cosines
2. Final Calculation
Find the length of side x in the triangle below. Round your final answer to one decimal place.
35° 80° 15 cm x
Show Your Setup
Final Answer
x = _________ cm
Self-Check
How confident do you feel identifying which law to use?
Not Yet Getting There I've Got This!
