What is the general solution for 4=tan^2(45+x/2) ?

The general solution for 4=tan^2(45+x/2) is x=2*1.10714…+360n-90,x=-2*1.10714…+360n-90

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4=tan^2(45+x/2)

Solution

4 = tan^{2} (4 5^{\circ} + \frac{x}{2})

Solution

x = 2 \cdot 1.10714 \dots + 36 0^{\circ} n - 9 0^{\circ}, x = - 2 \cdot 1.10714 \dots + 36 0^{\circ} n - 9 0^{\circ}

Radians

x = 2 \cdot 1.10714 \dots - \frac{π}{2} + 2 πn, x = - 2 \cdot 1.10714 \dots - \frac{π}{2} + 2 πn

Solution steps

4 = tan^{2} (4 5^{\circ} + \frac{x}{2})

Switch sides

tan^{2} (4 5^{\circ} + \frac{x}{2}) = 4

Solve by substitution

tan^{2} (4 5^{\circ} + \frac{x}{2}) = 4

Let:

tan (4 5^{\circ} + \frac{x}{2}) = u

u^{2} = 4

u^{2} = 4 : u = 2, u = - 2

u^{2} = 4

For

x^{2} = f (a)

the solutions are

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

u = 4, u = - 4

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

- 4 = - 2

- 4

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= - 2

u = 2, u = - 2

Substitute back

u = tan (4 5^{\circ} + \frac{x}{2})

tan (4 5^{\circ} + \frac{x}{2}) = 2, tan (4 5^{\circ} + \frac{x}{2}) = - 2

tan (4 5^{\circ} + \frac{x}{2}) = 2, tan (4 5^{\circ} + \frac{x}{2}) = - 2

tan (4 5^{\circ} + \frac{x}{2}) = 2 : x = 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

tan (4 5^{\circ} + \frac{x}{2}) = 2

Apply trig inverse properties

tan (4 5^{\circ} + \frac{x}{2}) = 2

General solutions for

tan (4 5^{\circ} + \frac{x}{2}) = 2

tan (x) = a \Rightarrow x = arctan (a) + 18 0^{\circ} n

4 5^{\circ} + \frac{x}{2} = arctan (2) + 18 0^{\circ} n

4 5^{\circ} + \frac{x}{2} = arctan (2) + 18 0^{\circ} n

Solve

4 5^{\circ} + \frac{x}{2} = arctan (2) + 18 0^{\circ} n : x = 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

4 5^{\circ} + \frac{x}{2} = arctan (2) + 18 0^{\circ} n

Move

4 5^{\circ}

to the right side

4 5^{\circ} + \frac{x}{2} = arctan (2) + 18 0^{\circ} n

Subtract

4 5^{\circ}

from both sides

4 5^{\circ} + \frac{x}{2} - 4 5^{\circ} = arctan (2) + 18 0^{\circ} n - 4 5^{\circ}

Simplify

\frac{x}{2} = arctan (2) + 18 0^{\circ} n - 4 5^{\circ}

\frac{x}{2} = arctan (2) + 18 0^{\circ} n - 4 5^{\circ}

Multiply both sides by

2

\frac{x}{2} = arctan (2) + 18 0^{\circ} n - 4 5^{\circ}

Multiply both sides by

2

\frac{2 x}{2} = 2 arctan (2) + 36 0^{\circ} n - 2 \cdot 4 5^{\circ}

Simplify

\frac{2 x}{2} = 2 arctan (2) + 36 0^{\circ} n - 2 \cdot 4 5^{\circ}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 arctan (2) + 36 0^{\circ} n - 2 \cdot 4 5^{\circ} : 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

2 arctan (2) + 36 0^{\circ} n - 2 \cdot 4 5^{\circ}

2 \cdot 4 5^{\circ} = 9 0^{\circ}

2 \cdot 4 5^{\circ}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= 9 0^{\circ}

Cancel the common factor:

2

= 9 0^{\circ}

= 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

x = 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

x = 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

x = 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

x = 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

tan (4 5^{\circ} + \frac{x}{2}) = - 2 : x = - 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

tan (4 5^{\circ} + \frac{x}{2}) = - 2

Apply trig inverse properties

tan (4 5^{\circ} + \frac{x}{2}) = - 2

General solutions for

tan (4 5^{\circ} + \frac{x}{2}) = - 2

tan (x) = - a \Rightarrow x = arctan (- a) + 18 0^{\circ} n

4 5^{\circ} + \frac{x}{2} = arctan (- 2) + 18 0^{\circ} n

4 5^{\circ} + \frac{x}{2} = arctan (- 2) + 18 0^{\circ} n

Solve

4 5^{\circ} + \frac{x}{2} = arctan (- 2) + 18 0^{\circ} n : x = - 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

4 5^{\circ} + \frac{x}{2} = arctan (- 2) + 18 0^{\circ} n

Simplify

arctan (- 2) + 18 0^{\circ} n : - arctan (2) + 18 0^{\circ} n

arctan (- 2) + 18 0^{\circ} n

Use the following property:

arctan (- x) = - arctan (x)

arctan (- 2) = - arctan (2)

= - arctan (2) + 18 0^{\circ} n

4 5^{\circ} + \frac{x}{2} = - arctan (2) + 18 0^{\circ} n

Move

4 5^{\circ}

to the right side

4 5^{\circ} + \frac{x}{2} = - arctan (2) + 18 0^{\circ} n

Subtract

4 5^{\circ}

from both sides

4 5^{\circ} + \frac{x}{2} - 4 5^{\circ} = - arctan (2) + 18 0^{\circ} n - 4 5^{\circ}

Simplify

\frac{x}{2} = - arctan (2) + 18 0^{\circ} n - 4 5^{\circ}

\frac{x}{2} = - arctan (2) + 18 0^{\circ} n - 4 5^{\circ}

Multiply both sides by

2

\frac{x}{2} = - arctan (2) + 18 0^{\circ} n - 4 5^{\circ}

Multiply both sides by

2

\frac{2 x}{2} = - 2 arctan (2) + 36 0^{\circ} n - 2 \cdot 4 5^{\circ}

Simplify

\frac{2 x}{2} = - 2 arctan (2) + 36 0^{\circ} n - 2 \cdot 4 5^{\circ}

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

- 2 arctan (2) + 36 0^{\circ} n - 2 \cdot 4 5^{\circ} : - 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

- 2 arctan (2) + 36 0^{\circ} n - 2 \cdot 4 5^{\circ}

2 \cdot 4 5^{\circ} = 9 0^{\circ}

2 \cdot 4 5^{\circ}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= 9 0^{\circ}

Cancel the common factor:

2

= 9 0^{\circ}

= - 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

x = - 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

x = - 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

x = - 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

x = - 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

Combine all the solutions

x = 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}, x = - 2 arctan (2) + 36 0^{\circ} n - 9 0^{\circ}

Show solutions in decimal form

x = 2 \cdot 1.10714 \dots + 36 0^{\circ} n - 9 0^{\circ}, x = - 2 \cdot 1.10714 \dots + 36 0^{\circ} n - 9 0^{\circ}

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Frequently Asked Questions (FAQ)

What is the general solution for 4=tan^2(45+x/2) ?
The general solution for 4=tan^2(45+x/2) is x=2*1.10714…+360n-90,x=-2*1.10714…+360n-90