Inverse tangents in cyclic order

In summary, the problem asks to find the value of tan(theta) if theta is equal to the inverse tangent of (a(a+b+c)/bc) plus the inverse tangent of (b(a+b+c)/ac) plus the inverse tangent of (c(a+b+c)/ab). Attempts at solving the problem involve considering the sides of a triangle and using their properties, but ultimately, the answer is found by using the equation for the sum of inverse tangents and solving for tan(theta) using the tangent sum formula.
  • #1
cr7einstein
87
2

Homework Statement



The problem- if

$$\theta= tan^{-1}(\frac{a(a+b+c)}{bc})+tan^{-1}(\frac{b(a+b+c)}{ac})+tan^{-1}(\frac{c(a+b+c)}{ab})$$
, then find $$tan\theta$$

Homework Equations

The Attempt at a Solution


I tried to use these as sides of a triangle and use their properties, but other than that I am clueless. I cannot think of a substitution either. The answer happens to be zero. Any help is appreciated. Thanks in advance!
 
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  • #2
Okay, I have it. Apparently, there is no 'elegant' way to do this. Just use the equation for $$arctan x + arctan y + arctan z$$, and after some really messy calculation, you get the answer as zero. If there is a different, neat answer, please let me know.
 
  • #3
i think your question is wrong it should be:-
in all the terms it should be sqrt(x) , sqrt(y), sqrt(z) inside of arctan ; where x,y,z are a(a+b+c)/bc , b(a+b+c)/ac , c(a+b+c)/ab ; as i have seen a similar question in my textbook.

if you consider it like this and take tan both sides and use tan(E+F+G)=(S1-S3)/(1-S2), it is easily solved in just few steps.
 

FAQ: Inverse tangents in cyclic order

1. What is the definition of inverse tangents in cyclic order?

Inverse tangents in cyclic order refers to the angular relationship between two points or objects in a circular motion, where the inverse tangent is used to calculate the angle between the two points. It is commonly used in trigonometry and geometry.

2. How is inverse tangents in cyclic order different from regular inverse tangents?

In regular inverse tangents, the input values are typically real numbers, while in inverse tangents in cyclic order, the input values are angles or points on a circle. Additionally, inverse tangents in cyclic order take into account the direction and order of the points on the circle, whereas regular inverse tangents do not.

3. What are some real-world applications of inverse tangents in cyclic order?

Inverse tangents in cyclic order are commonly used in navigation and robotics, where the angle between two points or objects on a circular path needs to be calculated. They are also used in astronomy to determine the positions of celestial bodies in relation to each other.

4. How do you calculate inverse tangents in cyclic order?

To calculate inverse tangents in cyclic order, you can use the inverse tangent function on a calculator or computer. Alternatively, you can use the formula arctan(y/x), where y and x are the coordinates of the two points on the circle.

5. Are there any limitations or drawbacks to using inverse tangents in cyclic order?

One limitation of inverse tangents in cyclic order is that they can only be used for objects or points that are on a circular path. Additionally, they may not accurately represent the true angle between two points if the circle is very large or if the points are close together. In these cases, other methods of calculating angles may be more appropriate.

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