Question 1

What is the Law of Cosines and when do you use it?

Accepted Answer

The Law of Cosines is a fundamental theorem in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. It's expressed as c² = a² + b² - 2ab×cos(C). Use it when you know: three sides (SSS), two sides and the included angle (SAS), or when the Law of Sines would result in an ambiguous case.

Question 2

How is the Law of Cosines different from the Law of Sines?

Accepted Answer

The Law of Cosines relates sides and angles through cosine functions and is best for SSS and SAS cases. The Law of Sines relates sides and angles through sine functions and works well for ASA, AAS, and some SSA cases. The Law of Cosines is more versatile but requires more complex calculations, while the Law of Sines is simpler but can have ambiguous solutions.

Question 3

What are the different triangle solving cases (SSS, SAS, ASA, etc.)?

Accepted Answer

Triangle solving cases are: SSS (three sides) - use Law of Cosines; SAS (two sides, included angle) - use Law of Cosines; ASA/AAS (two angles, one side) - use Law of Sines; SSA (two sides, non-included angle) - may have 0, 1, or 2 solutions. Our calculator handles all these cases automatically.

Question 4

How do you determine if a triangle is valid?

Accepted Answer

A triangle is valid if it satisfies the triangle inequality: the sum of any two sides must be greater than the third side. For angles, the sum must equal 180°. Additionally, each angle must be between 0° and 180°. Our calculator automatically checks these conditions and alerts you if the triangle is invalid.

Question 5

What's the difference between acute, right, and obtuse triangles?

Accepted Answer

Triangle classification is based on the largest angle: Acute triangles have all angles less than 90°; Right triangles have exactly one 90° angle; Obtuse triangles have one angle greater than 90°. You can also classify using sides: if c² < a² + b², it's acute; if c² = a² + b², it's right; if c² > a² + b², it's obtuse (where c is the longest side).

Question 6

How do you calculate the area of a triangle using the Law of Cosines?

Accepted Answer

Once you have all three sides using the Law of Cosines, you can calculate the area using Heron's formula: Area = √[s(s-a)(s-b)(s-c)], where s is the semiperimeter (a+b+c)/2. Alternatively, if you know two sides and the included angle, use Area = (1/2)ab×sin(C). Our calculator provides the area automatically using the most appropriate method.

Law of Cosines Calculator

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