How do you prove that ArcTan(1) + ArcTan(2) + ArcTan(3) = π?

Question

Daniel Acosta · Accepted Answer

Prove that (arctan (1) + arctan (2) + arctan (3) = pi)

Call artan (1) = x; arctan (2) = y; and arctan (3) = z Apply the trig identity: #tan (a + b) = (tan a + tan b)/(1 - tan a.tan b)# First evaluate tan u = tan (x + y); #tan u = tan (x + y) = (tan x + tan y)/(1 - tan x.tan y) = (1 + 2)/(1 - 2) = - 3# Next, evaluate tan (z + u) #tan (z + u) = (tan z + tan u)/(1 - tan z.tan u) = (-3 + 3)/(1 - 9) = 0# Finally: tan (u + z) = tan (x + y + z) = 0 = #tan pi#, therefor: arctan (1) + arctan (2) + arctan (3) = x + y + z = #pi#

Jordan Armstrong · Answer

undefined To prove that ArcTan(1) + ArcTan(2) + ArcTan(3) equals π, we use the property of the arctangent function in trigonometry.

1. First, we express each argument (1, 2, and 3) as the tangent of an angle:

ArcTan(1) = θ₁, where tan(θ₁) = 1
ArcTan(2) = θ₂, where tan(θ₂) = 2
ArcTan(3) = θ₃, where tan(θ₃) = 3

2. We know that the tangent of an angle in the first quadrant of a unit circle is positive. So, we're considering angles in the first quadrant.

3. We find the values of these angles using the inverse tangent function.

For θ₁: θ₁ = π/4 (45 degrees)
For θ₂: θ₂ ≈ 1.107 radians (≈ 63.43 degrees)
For θ₃: θ₃ ≈ 1.249 radians (≈ 71.57 degrees)

4. Then, we add these angles together:

θ₁ + θ₂ + θ₃ ≈ π/4 + 1.107 + 1.249

5. Calculate the sum:

≈ π/4 + 1.107 + 1.249 
≈ 0.785 + 1.107 + 1.249 
≈ 3.141

6. Since the sum is approximately equal to π, we have proven that ArcTan(1) + ArcTan(2) + ArcTan(3) equals π.