Just to prove them wrong: Trigonometry

In summary: Some sources use (-π/2, π/2) instead. So it is important to know which range is being used in order to accurately solve equations involving the arccotangent function.
  • #1
mooncrater
217
18

Homework Statement


There is a part of solution of a question which says that:
##cot|cot^{-1}x|=cot cot^{-1}x=x##

Homework Equations

The Attempt at a Solution


If we put ##x=-1## in this equation then ##cot^{-1}(-1)=-\pi /4## which will have a modulus =##\pi /4##. And ##cot \pi /4=1## which is not equal to the ##x## we took. So this equation seems wrong to me. Am I correct here?
 
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  • #2
You should know the domain and ranges.
What is the range for cot-1x ?
 
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  • #3
Raghav Gupta said:
You should know the domain and ranges.
What is the range for cot-1x ?
Hmmmm... I think I have lost it... it's domain is ## R## and has a range ## (0, \pi)##. So ##cot^{-1}(-1)=3\pi /4## not ##-\pi /4##.
I forgot its range...
So now this equation is true for all values of ##x##.
Thanks (##R\rightarrow R## help)
 
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  • #4
mooncrater said:

Homework Statement


There is a part of solution of a question which says that:
##cot|cot^{-1}x|=cot cot^{-1}x=x##

Homework Equations

The Attempt at a Solution


If we put ##x=-1## in this equation then ##cot^{-1}(-1)=-\pi /4## which will have a modulus =##\pi /4##. And ##cot \pi /4=1## which is not equal to the ##x## we took. So this equation seems wrong to me. Am I correct here?

If it is not equal to the x you took, it is equal to
| (the x you took) |
and there is a | | in the problem.

I think the answer you have been given is wrong and the answer is | x | .

You seem to have been given a few wrong answers and even wrong questions, or at least at the limit thereof. :oldwink:
 
  • #5
epenguin said:
If it is not equal to the x you took, it is equal to
| (the x you took) |
and there is a | | in the problem.

I think the answer you have been given is wrong and the answer is | x | .

You seem to have been given a few wrong answers and even wrong questions, or at least at the limit thereof. :oldwink:
The standard range used for the arccotangent is (0, π). Thus the arccotangent function always returns a positive value.

So we have | cot-1(x) | = cot-1(x)

Added in Edit:
However, the range used for the arccotangent function is not universally established.
 
Last edited:
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Related to Just to prove them wrong: Trigonometry

What is Trigonometry?

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is used to calculate and measure angles, distances, and heights, and has many practical applications in fields such as engineering and navigation.

Why is it important to "prove them wrong" in Trigonometry?

Proving someone wrong in Trigonometry means finding a solution or proof that contradicts their original belief or statement. This is important because it helps to strengthen our understanding and knowledge of the subject and allows us to discover new and more efficient ways of solving problems.

What are the main concepts in Trigonometry?

The main concepts in Trigonometry include angles, triangles, trigonometric functions (sine, cosine, tangent), inverse trigonometric functions, and the Pythagorean theorem. These concepts are used to solve various types of problems involving angles and distances.

How is Trigonometry used in everyday life?

Trigonometry has many practical applications in everyday life. It is used in architecture to design buildings and determine the height of structures. It is also used in navigation to calculate distances and angles. In addition, Trigonometry is used in fields such as astronomy, engineering, and physics.

What are some common misconceptions about Trigonometry?

One common misconception about Trigonometry is that it is only used in advanced mathematics and has no real-world applications. Another misconception is that it is a difficult subject to learn. In reality, Trigonometry has many real-world uses and can be easily understood with practice and proper instruction.

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