What is the general solution for arctan(w/2)+arctan(w/4)+arctan(w/6)=180 ?

The general solution for arctan(w/2)+arctan(w/4)+arctan(w/6)=180 is No Solution for w\in\mathbb{R}

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arctan(w/2)+arctan(w/4)+arctan(w/6)=180

arctan (\frac{w}{2}) + arctan (\frac{w}{4}) + arctan (\frac{w}{6}) = 18 0^{\circ}

No Solution for w \in R

Solution steps

arctan (\frac{w}{2}) + arctan (\frac{w}{4}) + arctan (\frac{w}{6}) = 18 0^{\circ}

Subtract

18 0^{\circ}

from both sides

arctan (\frac{w}{2}) + arctan (\frac{w}{4}) + arctan (\frac{w}{6}) - 18 0^{\circ} = 0

Rewrite using trig identities

- 18 0^{\circ} + arctan (\frac{\frac{6 w}{8 - w ^{2}} + \frac{w}{6}}{1 - \frac{6 w}{8 - w ^{2}} \cdot \frac{w}{6}}) = 0

Move

18 0^{\circ}

to the right side

arctan (\frac{\frac{6 w}{8 - w ^{2}} + \frac{w}{6}}{1 - \frac{6 w}{8 - w ^{2}} \cdot \frac{w}{6}}) = 18 0^{\circ}

- \frac{π}{2} < arctan (x) < \frac{π}{2}

No Solution

Verify solutions by plugging them into the original equation

No Solution for w \in R

sin (x + \frac{π}{6}) = 0

7 sin (x) + 3 = sin (x) + 2

24 cos (x) = 122

cos (3 x) = \frac{3}{3}

- 6 csc^{2} (x) + cot (x) = - 18

What is the general solution for arctan(w/2)+arctan(w/4)+arctan(w/6)=180 ?
The general solution for arctan(w/2)+arctan(w/4)+arctan(w/6)=180 is No Solution for w\in\mathbb{R}