Law of Cosines Game

Law of Cosines Game
The Law of Cosines is rule in trigonometry for solving oblique triangles (triangles without a right angle). The Law of Cosines is used to solve triangles in two main scenarios:

1. SAS (Side-Angle-Side): You know two sides and the angle between them (the included angle). This is used to find the third side.
   
2. SSS (Side-Side-Side): You know all three sides. This is used to find any of the three angles.
   Scroll down the page for a more detailed explanation.
    
   This game helps you practice finding missing sides and angles in non-right triangles using the Law of Cosines. You will be required to either find the side given SAS or find the angle given SSS. Calculate and select one of the answers. It includes a scoring system, and clear feedback to help you master this skill.
    
   

 

How to Play the Law of Cosines Game
This game focuses on non-right triangles and requires applying the Law of Cosines.
Here’s how to play:

1. Select a Practice Mission:
   Module 01 (SAS): Solve for missing side length c given two sides and the included angle (a, b, C).
   Module 02 (SSS): Solve for an unknown interior angle C given all three sides (a, b, c).
   Module 03 (Real-World Applications): Apply formulas to real-world scenarios across navigation, engineering, aviation, surveying, and astronomy.
   Module 04 (Mixed Challenge Quest): Test total mastery across all question types in a randomized set.
   
2. Analyze the Given Values: Read the problem stem and inspect the given side lengths or angles displayed in the target prompt box.
   
3. Select Your Answer: Choose the correct calculated value from the four multiple-choice options before moving to the next item.
   
4. Review Detailed Explanations: After each selection, review the step-by-step mathematical breakdown showing substitution and intermediate calculation steps.
   
5. Track Progress: Earn points for each correct response, toggle audio cues on or off as needed, and track performance across 10 rounds per session.
    
   

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The Law of Cosines
The Law of Cosines is used to solve triangles in two main scenarios:

1. SAS (Side-Angle-Side): You know two sides and the angle between them (the included angle). This is used to find the third side.
   
2. SSS (Side-Side-Side): You know all three sides. This is used to find any of the three angles.
    
   The Formulas
   For a triangle with angles \(A\), \(B\), and \(C\), and sides \(a\), \(b\), and \(c\) opposite those angles, the Law of Cosines is stated as:
   To Find a Side (SAS Case)
   The formula is set up to find the side opposite the known angle.
   \(a^2 = b^2 + c^2 - 2bc \cos(A)\)
   \(b^2 = a^2 + c^2 - 2ac \cos(B)\)
   \(c^2 = a^2 + b^2 - 2ab \cos(C)\)
   
   To Find an Angle (SSS Case)
   These formulas are derived by isolating the \(\cos\) term in the side formulas.
   \(\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}\)
   \(\cos(B) = \frac{a^2 + c^2 - b^2}{2ac}\)
   \(\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}\)
    
   Using the Law of Cosines to Find a Side (SAS)
   Goal: Find side \(a\), given side \(b\), side \(c\), and the included angle A.
   Example:
   A triangle has side \(b = 10\), side \(c = 15\), and the included angle \(A = 35^\circ\). Find side \(a\).
   Step-by-Step Solution:
   Choose the appropriate formula. Since we know \(b\), \(c\), and \(a\), we use the formula for \(a^2\):
   \(a^2 = b^2 + c^2 - 2bc \cos(A)\)
   Substitute the known values:
   \(a^2 = (10)^2 + (15)^2 - 2(10)(15) \cos(35^\circ)\)
   Calculate the squares and the product term:
   \(a^2 = 100 + 225 - 300 \cdot \cos(35^\circ)\)
   Find the cosine value:
   \(\cos(35^\circ) \approx 0.8192\)
   Calculate \(a^2\):
   \(a^2 = 325 - 300(0.8192)\)
   \(a^2 = 325 - 245.76\)
   \(a^2 = 79.24\)
   Take the square root to find \(a\):
   \(a = \sqrt{79.24} \approx 8.90\)
   The length of side \(a\) is approximately \(8.90\) units.
    
   Using the Law of Cosines to Find an Angle (SSS)
   Goal: Find angle \(C\), given side \(a\), side \(b\), and side \(c\).
   Example:
   A triangle has side \(a = 7\), side \(b = 9\), and side \(c = 12\). Find angle \(c\).
   Step-by-Step Solution:
   Choose the appropriate angle formula. Since we are looking for angle \(c\), we use the formula:
   \(\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}\)
   Substitute the known values:
   \(\cos(C) = \frac{(7)^2 + (9)^2 - (12)^2}{2(7)(9)}\)
   Calculate the squares and products:
   \(\cos(C) = \frac{49 + 81 - 144}{126}\)
   Simplify the numerator and denominator:
   \(\cos(C) = \frac{130 - 144}{126}\)
   \(\cos(C) = \frac{-14}{126} \approx -0.1111\)
   Use the inverse cosine function (\(\arccos\) or \(\cos^{-1}\)) to find the angle:
   \(C = \arccos(-0.1111)\)
   \(C \approx 96.38^\circ\)
   The measure of angle C is approximately \(96.38^\circ\). (Note: Since \(\cos(C)\) is negative, the angle is obtuse, which is correct for this geometry).
    
   

Common Student Misconceptions to Address

1. Order of Operations Error in SAS: Students frequently calculate a² + b² - 2ab prior to multiplying by \cos C. Remind students that multiplication takes precedence over subtraction:
   (a² + b²) - (2ab · cos C) NOT (a² + b² - 2ab) · cos C
   
2. Incorrect Angle Identification in SSS: When calculating angle C, students sometimes subtract the wrong side squared. Emphasize that the side being subtracted in the numerator (c²) must always be opposite the target angle (C).
   
3. Degree vs. Radian Mode: Remind students to verify their physical scientific calculators are set to Degree Mode (DEG) when following along with the game’s calculations.
   

The video gives a clear, step-by-step approach to learn how to find angles and sides using the Law of Cosines.

• Show Video

 

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