What is the general solution for arctan(x)+arctan(3x)+arctan(7x)=180 ?

The general solution for arctan(x)+arctan(3x)+arctan(7x)=180 is No Solution for x\in\mathbb{R}

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arctan(x)+arctan(3x)+arctan(7x)=180

arctan (x) + arctan (3 x) + arctan (7 x) = 18 0^{\circ}

No Solution for x \in R

Solution steps

arctan (x) + arctan (3 x) + arctan (7 x) = 18 0^{\circ}

Subtract

18 0^{\circ}

from both sides

arctan (x) + arctan (3 x) + arctan (7 x) - 18 0^{\circ} = 0

Rewrite using trig identities

arctan (3 x) + arctan (7 x) + arctan (x) - 18 0^{\circ}

Use the Sum to Product identity:

arctan (s) + arctan (t) = arctan (\frac{s + t}{1 - s t})

= arctan (x) - 18 0^{\circ} + arctan (\frac{3 x + 7 x}{1 - 3 x \cdot 7 x})

\frac{3 x + 7 x}{1 - 3 x \cdot 7 x} = \frac{10 x}{1 - 21 x ^{2}}

\frac{3 x + 7 x}{1 - 3 x \cdot 7 x}

Add similar elements:

3 x + 7 x = 10 x

= \frac{10 x}{1 - 3 \cdot 7 xx}

1 - 3 x \cdot 7 x = 1 - 21 x^{2}

1 - 3 x \cdot 7 x

3 x \cdot 7 x = 21 x^{2}

3 x \cdot 7 x

Multiply the numbers:

3 \cdot 7 = 21

= 21 xx

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

xx = x^{1 + 1}

= 21 x^{1 + 1}

Add the numbers:

1 + 1 = 2

= 21 x^{2}

= 1 - 21 x^{2}

= \frac{10 x}{1 - 21 x ^{2}}

= arctan (x) - 18 0^{\circ} + arctan (\frac{10 x}{1 - 21 x ^{2}})

Use the Sum to Product identity:

arctan (s) + arctan (t) = arctan (\frac{s + t}{1 - s t})

= - 18 0^{\circ} + arctan (\frac{\frac{10 x}{1 - 21 x ^{2}} + x}{1 - \frac{10 x}{1 - 21 x ^{2}} x})

- 18 0^{\circ} + arctan (\frac{\frac{10 x}{1 - 21 x ^{2}} + x}{1 - \frac{10 x}{1 - 21 x ^{2}} x}) = 0

Move

18 0^{\circ}

to the right side

- 18 0^{\circ} + arctan (\frac{\frac{10 x}{1 - 21 x ^{2}} + x}{1 - \frac{10 x}{1 - 21 x ^{2}} x}) = 0

Add

18 0^{\circ}

to both sides

- 18 0^{\circ} + arctan (\frac{\frac{10 x}{1 - 21 x ^{2}} + x}{1 - \frac{10 x}{1 - 21 x ^{2}} x}) + 18 0^{\circ} = 0 + 18 0^{\circ}

Simplify

arctan (\frac{\frac{10 x}{1 - 21 x ^{2}} + x}{1 - \frac{10 x}{1 - 21 x ^{2}} x}) = 18 0^{\circ}

arctan (\frac{\frac{10 x}{1 - 21 x ^{2}} + x}{1 - \frac{10 x}{1 - 21 x ^{2}} x}) = 18 0^{\circ}

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

No Solution

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

arctan (x) + arctan (3 x) + arctan (7 x) = 18 0^{\circ}

Remove the ones that don't agree with the equation.

No Solution for x \in R

2 cos (a) = 1

cos^{2} (x) = (\frac{1}{2})

3 sin (\frac{π}{2} x) = 2

3 cos (A) - 1 = 0

cot (\frac{x}{2}) = - 1

What is the general solution for arctan(x)+arctan(3x)+arctan(7x)=180 ?
The general solution for arctan(x)+arctan(3x)+arctan(7x)=180 is No Solution for x\in\mathbb{R}