What is the general solution for tan^2(x)= 1/2 ?

The general solution for tan^2(x)= 1/2 is x=0.61547…+pin,x=-0.61547…+pin

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tan^2(x)= 1/2

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

x = 0.61547 \dots + πn, x = - 0.61547 \dots + πn

Degrees

x = 35.26438 \dots^{\circ} + 18 0^{\circ} n, x = - 35.26438 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

tan (x) = u

u^{2} = \frac{1}{2}

u^{2} = \frac{1}{2} : u = \frac{1}{2}, u = - \frac{1}{2}

u^{2} = \frac{1}{2}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{1}{2}, u = - \frac{1}{2}

Substitute back

u = tan (x)

tan (x) = \frac{1}{2}, tan (x) = - \frac{1}{2}

tan (x) = \frac{1}{2}, tan (x) = - \frac{1}{2}

tan (x) = \frac{1}{2} : x = arctan (\frac{1}{2}) + πn

tan (x) = \frac{1}{2}

Apply trig inverse properties

tan (x) = \frac{1}{2}

General solutions for

tan (x) = \frac{1}{2}

tan (x) = a \Rightarrow x = arctan (a) + πn

x = arctan (\frac{1}{2}) + πn

x = arctan (\frac{1}{2}) + πn

tan (x) = - \frac{1}{2} : x = arctan (- \frac{1}{2}) + πn

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

Apply trig inverse properties

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

General solutions for

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

tan (x) = - a \Rightarrow x = arctan (- a) + πn

x = arctan (- \frac{1}{2}) + πn

x = arctan (- \frac{1}{2}) + πn

Combine all the solutions

x = arctan (\frac{1}{2}) + πn, x = arctan (- \frac{1}{2}) + πn

Show solutions in decimal form

x = 0.61547 \dots + πn, x = - 0.61547 \dots + πn

3 cos (x) - sin (2 x) = 0

5 cos^{2} (x) + 3 cos (x) = 0

- \frac{2}{3} cos (3 θ) = 1

x, y = sin (x y)

tan (t) = \frac{1}{3}

What is the general solution for tan^2(x)= 1/2 ?
The general solution for tan^2(x)= 1/2 is x=0.61547…+pin,x=-0.61547…+pin