Proving tan^-1(4/3)=2tan^-1(0.5)

  • Thread starter inv
  • Start date
In summary, by considering the argument of u/u*, it can be proven that tan^-1(4/3)=2tan^-1(0.5). This can be achieved by observing that arg u and arg u* are the same, which simplifies the equation to 2 tan A = 1. From there, using the equation tan2A=2tanA/(1-tan^2 A), it can be shown that tan2A(1-tan^2 A)=1, leading to the desired result. Additionally, drawing the complex numbers on an argand diagram can aid in understanding the concept.
  • #1
inv
46
0
[Solved]u=2+i.By considering the arg of u/u* or otherwise

*Solved


Homework Statement


u=2+i.
u*=2-i
"By considering the arg of u"/u* or otherwise prove that tan^-1(4/3)=2tan^-1(0.5).That's the question,explain how do pls?

Homework Equations


arg(u/u*)=arg u -arg u*
tan A=opposite side/adjacent side
tan2A=2tanA/(1-tan^2 A)

The Attempt at a Solution


tanA=1/2
2tanA=1
tan2A(1-tan^2 A)=1
*Stuck here,guide pls?

Edit*
 
Last edited:
Physics news on Phys.org
  • #2
draw you complex numbers on an argand diagram
observe that arg u and arg u* are the same... arg u + arg u* = arg u - (-arg u) = 2 arg u
and that tan^-1 x gives you an angle.
 
  • #3


I would approach this problem using trigonometric identities and properties. Firstly, we can rewrite the given equation as tan^-1(4/3) = 2tan^-1(1/2). Next, we can use the double angle formula for tangent, which states that tan2A = 2tanA/(1-tan^2 A). In this case, A is equal to tan^-1(1/2), so we can substitute it into the formula as follows:

tan2(tan^-1(1/2)) = 2tan(tan^-1(1/2))/(1-tan^2(tan^-1(1/2)))

Using the property that tan^-1(x) is the inverse of tan(x), we can simplify this to:

tan2(tan^-1(1/2)) = 2(1/2)/(1-(1/2)^2)

tan2(tan^-1(1/2)) = 1

Now, we can use the inverse tangent function to find the angle that has a tangent of 1:

tan^-1(1) = π/4

Therefore, we have shown that tan^-1(4/3) = 2tan^-1(1/2) = 2π/4 = π/2.

In terms of the argument of u/u*, we can interpret this as finding the angle of the complex number u divided by its complex conjugate u*. The argument of a complex number is the angle it makes with the positive real axis. In this case, u=2+i would have an argument of tan^-1(1/2) because it forms a right triangle with sides of 1 and 2, and the tangent of the angle opposite the side with length 1 is 1/2. Similarly, u*=2-i would have an argument of -tan^-1(1/2). Therefore, the argument of u/u* would be equal to tan^-1(1/2)-(-tan^-1(1/2)) = 2tan^-1(1/2).

In conclusion, using trigonometric identities and properties, we have shown that tan^-1(4/3)=2tan^-1(1/2), which can also be interpreted in terms of the arguments of complex numbers.
 

1. How do you prove tan^-1(4/3)=2tan^-1(0.5)?

To prove this statement, we will use the tangent addition formula: tan(a+b) = (tan(a) + tan(b)) / (1 - tan(a)tan(b)). First, let a = tan^-1(0.5) and b = tan^-1(4/3). Substituting these values into the formula, we get:
tan(tan^-1(0.5) + tan^-1(4/3)) = (tan(tan^-1(0.5)) + tan(tan^-1(4/3))) / (1 - tan(tan^-1(0.5))tan(tan^-1(4/3)))
Simplifying this, we get:
tan(tan^-1(0.5) + tan^-1(4/3)) = (0.5 + 4/3) / (1 - 0.5 * 4/3)
tan(tan^-1(0.5) + tan^-1(4/3)) = 2
Since we know that tan(a+b) = tan(2tan^-1(0.5)), we can conclude that tan^-1(4/3) = 2tan^-1(0.5).

2. Can you prove tan^-1(4/3)=2tan^-1(0.5) without using the tangent addition formula?

Yes, there are other ways to prove this statement. One way is by using the inverse trigonometric identity: tan^-1(x) = arctan(x). Using this identity, we can rewrite the statement as:
arctan(4/3) = 2arctan(0.5)
Next, we can use the double angle formula for arctan:
arctan(2x / (1-x^2)) = 2arctan(x)
Substituting x = 0.5, we get:
arctan(2(0.5) / (1-(0.5)^2)) = 2arctan(0.5)
Simplifying this, we get:
arctan(4/3) = 2arctan(0.5)
Therefore, we have proven that tan^-1(4/3)=2tan^-1(0.5).

3. What is the significance of proving tan^-1(4/3)=2tan^-1(0.5)?

This proof shows that the inverse tangent function is not linear, meaning that the angle does not double when the input is doubled. This is important because it helps us understand the behavior of trigonometric functions and their inverses, and it allows us to solve more complex problems involving trigonometry.

4. Can this statement be proven using geometric reasoning?

Yes, this statement can also be proven using geometric reasoning. We can draw a right triangle with sides 4, 3, and 5, and label one of the acute angles as theta. Then, using the definition of tangent as opposite over adjacent, we can see that tan(theta) = 3/4.
Next, we can draw a smaller right triangle within the original triangle, with sides 1, 0.5, and √1.25. Labeling the same acute angle theta, we can see that tan(theta) = 0.5/1 = 0.5.
Using the fact that the two triangles are similar, we can see that the ratio of the adjacent sides (1/4) is the same as the ratio of the opposite sides (0.5/3). This means that the angles in both triangles must be equal, so tan^-1(4/3) = tan^-1(0.5).
Finally, since we know that tan^-1(x) is the angle whose tangent is x, we can conclude that tan^-1(4/3) = 2tan^-1(0.5).

5. Are there any practical applications of this proof?

Yes, this proof has practical applications in fields such as engineering, physics, and navigation. For example, in engineering and physics, trigonometric functions are often used to model and solve real-world problems, and understanding their properties is crucial for accurate calculations. In navigation, the inverse tangent function is

Similar threads

  • Precalculus Mathematics Homework Help
Replies
5
Views
524
  • Precalculus Mathematics Homework Help
Replies
6
Views
2K
  • Precalculus Mathematics Homework Help
Replies
1
Views
1K
  • Precalculus Mathematics Homework Help
Replies
1
Views
936
  • Precalculus Mathematics Homework Help
Replies
2
Views
2K
  • Precalculus Mathematics Homework Help
Replies
1
Views
4K
  • Precalculus Mathematics Homework Help
Replies
10
Views
3K
  • Precalculus Mathematics Homework Help
Replies
15
Views
2K
  • Precalculus Mathematics Homework Help
Replies
13
Views
2K
  • Precalculus Mathematics Homework Help
Replies
5
Views
993
Back
Top