Finding the Value of arctan(1) Without a Calculator

  • Thread starter kasse
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In summary, the arctan function has a domain of the whole real line and a range of values of (-0.5*pi, 0.5*pi). The value of arctan(1) is equivalent to the angle whose tangent is 1. This can be found by reversing the unit circle and looking for the angle that matches the given value. The equation tan(x)=1 can also be used to solve for the value of arctan(1) by taking the inverse tangent.
  • #1
kasse
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All I know about the arctan function is that its domain of def. is the whole real line and that the range of values is (-0,5*pi , 0,5*pi), and also that arctan(0)=0.

But is there a way to know that arctan(1)=(1/4)*pi without recognizing the decimal number the calculator gives?
 
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  • #3
I think I figured out why after some brainwork:

1=tan (arctan (1))

so because tan(pi/4)=1, arctan(1)=pi/4
 
Last edited:
  • #4
i don't know if this will help but take the unit circle and reverse it. instead of looking for the angle then finding the value. look for the value then look for the angle that gives you the value. for arcsin(1), you would go along looking a the value of tan(angle) then when you find the value, the x in arcsin(x), you look at the angle that matches up with it.
 
  • #5
That's easy. Just denote [itex] \arctan 1 =x [/itex] and apply the [itex] \tan [/itex] on both members of the equation. You'll find the eqn [itex] \tan x =1 [/itex] which can easily be solved.

Daniel.
 
  • #6
The simplest definition of [itex]tan(\theta)[/itex] is "opposite side divided by near side" in a right triangle. If [itex]tan(\theta)= 1[/itex] then the two legs of the right triangle are the same length- it is an isosceles right triangle. What does that tell you about the angles?
 

1. What is arctan(1)?

arctan(1) refers to the inverse tangent function of 1. It finds the angle whose tangent is 1. In trigonometry, this function is used to determine an angle given the ratio of the opposite side to the adjacent side in a right triangle.

2. What is the Value of arctan(1)?

The value of arctan(1) is π/4 or 45 degrees. This is because in a right triangle, if the lengths of the opposite and adjacent sides are equal, the tangent of the angle is 1, and the angle is 45 degrees or π/4 radians.

3. Can You Compute arctan(1) Using a Calculator?

Yes, arctan(1) can be easily computed using a scientific calculator. Most calculators have an 'arctan' or 'tan⁻¹' function. By entering 1 into this function, the calculator will return π/4 or 45 degrees, depending on its mode (radians or degrees).

4. How is arctan(1) Used in Trigonometry?

In trigonometry, arctan(1) is used in solving problems involving right triangles where the lengths of two sides are known. It is also used in various trigonometric identities and equations, especially in problems involving circular motion and waves.

5. Is There a Series Expansion for arctan(1)?

Yes, arctan(x) can be expanded as a Taylor Series, and for arctan(1), the series is: arctan(1) = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ... This is an alternating series and converges to π/4.

6. Are There Any Geometric Methods to Compute arctan(1)?

Geometrically, arctan(1) can be visualized using a unit circle. Since the tangent of an angle in a unit circle is the y-coordinate divided by the x-coordinate, an angle whose tangent is 1 will have equal x and y coordinates. This occurs at an angle of 45 degrees or π/4 radians.

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