What Is a Free Law of Cosines Calculator?
A Free Law of Cosines Calculator is an online mathematics tool that helps calculate missing side lengths or angles in a triangle when enough information is known. It is especially useful for triangles that are not right triangles.
The Law of Cosines extends the Pythagorean theorem to any triangle.
c2=a2+b2−2abcos(C)c^2 = a^2 + b^2 – 2ab\cos(C)
c=52+62−2(5)(6)cos(120∘)=9.54c=\sqrt{\text{5}^2+\text{6}^2-2(\text{5})(\text{6})\cos(\text{120}^\circ)}=\text{9.54}c=52+62−2(5)(6)cos(120∘)=9.54
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aaa
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CCCC = 120°a = 5b = 6c = 9.5
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Law of Cosines Formula
For a triangle with sides aa, bb, and cc, and angle CC opposite side cc: c2=a2+b2−2abcos(C)c^2=a^2+b^2-2ab\cos(C)
The formula can also be rearranged to find an angle: cos(C)=a2+b2−c22ab\cos(C)=\frac{a^2+b^2-c^2}{2ab}
Then: C=cos−1(a2+b2−c22ab)C=\cos^{-1}\left(\frac{a^2+b^2-c^2}{2ab}\right)
Example
Suppose a triangle has:
- a=5a=5
- b=7b=7
- C=60∘C=60^\circ
Using the Law of Cosines: c2=52+72−2(5)(7)cos(60∘)c^2=5^2+7^2-2(5)(7)\cos(60^\circ) c2=25+49−70(0.5)c^2=25+49-70(0.5) c2=39c^2=39
Therefore: c=39≈6.245c=\sqrt{39}\approx6.245
The missing side is approximately 6.245 units.
How a Free Law of Cosines Calculator Works
A typical calculator allows you to:
- Enter two known side lengths and the included angle.
- Calculate the missing side.
- Or enter three side lengths to calculate an angle.
- Select degrees or radians when appropriate.
- Instantly receive the result.
When Is the Law of Cosines Useful?
A Free Law of Cosines Calculator can help with:
- Geometry homework
- Trigonometry
- Triangle calculations
- Surveying
- Navigation
- Engineering
- Construction
- Physics
- Finding unknown sides
- Finding unknown angles
- Checking manual calculations
Law of Cosines vs. Pythagorean Theorem
The Pythagorean theorem applies specifically to right triangles: a2+b2=c2a^2+b^2=c^2
The Law of Cosines works for general triangles: c2=a2+b2−2abcos(C)c^2=a^2+b^2-2ab\cos(C)
When C=90∘C=90^\circ, the cosine term becomes zero, and the Law of Cosines becomes the Pythagorean theorem.
In simple terms: a Free Law of Cosines Calculator quickly finds missing sides or angles of a triangle using the Law of Cosines, making complex triangle calculations much easier.
Explore the Law of Cosines
