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.
    
   

Law of Cosines

Progress Score: 0

SAS Case c² = a² + b² - 2ab cos(C)

Find side c

Next Problem ➜

 

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. Start: Each Quiz consists of 10 questions. Click “Start Game”.
   
2. Look at the Problem: You will be required to either find the side given SAS or find the angle given SSS.
   
3. Select Your Answer: Calculate and select you answer.br>
4. Check Your Work: The game will tell you if you’re correct. If you are wrong, the correct answer will be highlighted in green.
   
5. Get a New Problem: Click “Next Problem” for a new problem.
   Your score is tracked, showing how many you’ve gotten right.
   
6. Finish Game When you have completed 10 questions, the game will give you your final score.
    
   

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).
    
   

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

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