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Homework Statement
In any triangle ABC, prove that $$ a^2 + b^2 +c^2 =4 \Delta (\cot {A}+\cot {B}+\cot {C}) $$
Wrichik Basu said:Homework Statement
In any triangle ABC, prove that $$ a^2 + b^2 +c^2 =4 \Delta (\cot {A}+\cot {B}+\cot {C}) $$
Homework Equations
The Attempt at a Solution
View attachment 203380
Buffu said:What is ##\Delta## ?
Wrichik Basu said:The area of Triangle, the standard notation used in Trigonometry.
Buffu said:##\displaystyle {c \over \sin^2 c} = 2R##
Check your work again, you replace ##\sin c## with ##a/(2R)##, you should do ##c/(2R)##
Yes, that was a typo.Wrichik Basu said:You are wrong: $$\frac {c} {\sin (c)} = 2R$$ not $$\frac {c}{\sin ^2 (c)} = 2R$$
Buffu said:What is Δ ?
No, this is not standard notation for the area of a triangle. Δ is sometimes used for the discriminant of a quadratic equation, but I have never seen it used for area of any kind.Wrichik Basu said:The area of Triangle, the standard notation used in Trigonometry.
Mark44 said:No, this is not standard notation for the area of a triangle. Δ is sometimes used for the discriminant of a quadratic equation, but I have never seen it used for area of any kind.
The basic properties of triangles in trigonometry include the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Other important properties include the law of sines and the law of cosines, which can be used to solve for the unknown sides and angles of a triangle.
A triangle is considered a right triangle if it has one angle that measures exactly 90 degrees. This can be determined by using the Pythagorean theorem or by checking if the sides are in a 3-4-5 ratio, which is a common right triangle ratio. Additionally, if two sides of a triangle are perpendicular, it is also a right triangle.
The law of sines is used to solve for missing sides or angles in any triangle, while the law of cosines is specifically used to solve for missing sides or angles in a triangle with a known angle and two known sides. The law of sines uses ratios of the side lengths to the sine of the corresponding angle, while the law of cosines uses the cosine of an angle to solve for a side length.
Trigonometry properties, such as the Pythagorean theorem and the laws of sines and cosines, can be applied to real-world problems involving angles and distances. For example, they can be used in navigation and surveying to determine the height of a building or the distance between two points. They can also be used in engineering and construction to calculate the lengths of sides and angles in a structure.
Some common mistakes to avoid when using trigonometry properties include using the wrong formula for the given problem, forgetting to convert units of measurement, and not properly labeling the sides and angles of a triangle. It is important to double check all steps and calculations to ensure accuracy in solving the problem.