The Law of Cosines Explained
Your complete guide for any triangle!
What is the Law of Cosines? (For Everyone!)
Imagine a triangle that is NOT a right-angled triangle. How can you find its sides and angles? The Law of Cosines is a super useful rule in geometry that helps us with that! It's like an advanced version of the Pythagorean Theorem that works for any triangle.
The Cosine Rule Formulas
For any triangle with sides a, b, c and opposite angles A, B, C, the Law of Cosines states:
To find a side:
$$ a^2 = b^2 + c^2 - 2bc \cos(A) $$ $$ b^2 = a^2 + c^2 - 2ac \cos(B) $$ $$ c^2 = a^2 + b^2 - 2ab \cos(C) $$To find an angle (just rearranged):
$$ \cos(A) = \frac{b^2 + c^2 - a^2}{2bc} $$When Should You Use It?
The Law of Cosines is your best friend in two main situations:
- Side-Angle-Side (SAS): You know the lengths of two sides and the angle between them. You want to find the length of the third side.
- Side-Side-Side (SSS): You know the lengths of all three sides. You want to find the measure of any of the three angles.
Example 1: Finding a Missing Side (SAS)
Let's say we have a triangle where side b = 10, side c = 12, and the angle between them is A = 40°. We want to find side a.
We use the formula: \( a^2 = b^2 + c^2 - 2bc \cos(A) \)
- Plug in the values: \( a^2 = 10^2 + 12^2 - 2(10)(12) \cos(40°) \)
- Calculate the squares: \( a^2 = 100 + 144 - 240 \cos(40°) \)
- Find the cosine value: \( \cos(40°) \approx 0.766 \)
- Multiply: \( a^2 = 244 - 240(0.766) \approx 244 - 183.85 \)
- Subtract: \( a^2 \approx 60.15 \)
- Take the square root: \( a \approx \sqrt{60.15} \approx 7.76 \)
So, the missing side a is about 7.76 units long.
Example 2: Finding a Missing Angle (SSS)
Let's say we have a triangle with sides a = 8, b = 10, and c = 12. We want to find angle A.
We use the formula: \( \cos(A) = \frac{b^2 + c^2 - a^2}{2bc} \)
- Plug in the values: \( \cos(A) = \frac{10^2 + 12^2 - 8^2}{2(10)(12)} \)
- Calculate the squares: \( \cos(A) = \frac{100 + 144 - 64}{240} \)
- Simplify the numerator: \( \cos(A) = \frac{180}{240} \)
- Divide: \( \cos(A) = 0.75 \)
- Find the inverse cosine: \( A = \cos^{-1}(0.75) \approx 41.4° \)
So, the missing angle A is about 41.4 degrees.
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