What is the general solution for tan^2(45+x/3)=0.58 ?

The general solution for tan^2(45+x/3)=0.58 is x=3*0.65086…+540n-135,x=-3*0.65086…+540n-135

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tan^2(45+x/3)=0.58

Solution

tan^{2} (4 5^{\circ} + \frac{x}{3}) = 0.58

Solution

x = 3 \cdot 0.65086 \dots + 54 0^{\circ} n - 13 5^{\circ}, x = - 3 \cdot 0.65086 \dots + 54 0^{\circ} n - 13 5^{\circ}

Radians

x = 3 \cdot 0.65086 \dots - \frac{3 π}{4} + 3 πn, x = - 3 \cdot 0.65086 \dots - \frac{3 π}{4} + 3 πn

Solution steps

tan^{2} (4 5^{\circ} + \frac{x}{3}) = 0.58

Solve by substitution

tan^{2} (4 5^{\circ} + \frac{x}{3}) = 0.58

Let:

tan (4 5^{\circ} + \frac{x}{3}) = u

u^{2} = 0.58

u^{2} = 0.58 : u = 0.58, u = - 0.58

u^{2} = 0.58

For

x^{2} = f (a)

the solutions are

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

u = 0.58, u = - 0.58

Substitute back

u = tan (4 5^{\circ} + \frac{x}{3})

tan (4 5^{\circ} + \frac{x}{3}) = 0.58, tan (4 5^{\circ} + \frac{x}{3}) = - 0.58

tan (4 5^{\circ} + \frac{x}{3}) = 0.58, tan (4 5^{\circ} + \frac{x}{3}) = - 0.58

tan (4 5^{\circ} + \frac{x}{3}) = 0.58 : x = 3 arctan (0.58) + 54 0^{\circ} n - 13 5^{\circ}

tan (4 5^{\circ} + \frac{x}{3}) = 0.58

Apply trig inverse properties

tan (4 5^{\circ} + \frac{x}{3}) = 0.58

General solutions for

tan (4 5^{\circ} + \frac{x}{3}) = 0.58

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

4 5^{\circ} + \frac{x}{3} = arctan (0.58) + 18 0^{\circ} n

4 5^{\circ} + \frac{x}{3} = arctan (0.58) + 18 0^{\circ} n

Solve

4 5^{\circ} + \frac{x}{3} = arctan (0.58) + 18 0^{\circ} n : x = 3 arctan (0.58) + 54 0^{\circ} n - 13 5^{\circ}

4 5^{\circ} + \frac{x}{3} = arctan (0.58) + 18 0^{\circ} n

Move

4 5^{\circ}

to the right side

4 5^{\circ} + \frac{x}{3} = arctan (0.58) + 18 0^{\circ} n

Subtract

4 5^{\circ}

from both sides

4 5^{\circ} + \frac{x}{3} - 4 5^{\circ} = arctan (0.58) + 18 0^{\circ} n - 4 5^{\circ}

Simplify

\frac{x}{3} = arctan (0.58) + 18 0^{\circ} n - 4 5^{\circ}

\frac{x}{3} = arctan (0.58) + 18 0^{\circ} n - 4 5^{\circ}

Multiply both sides by

3

\frac{x}{3} = arctan (0.58) + 18 0^{\circ} n - 4 5^{\circ}

Multiply both sides by

3

\frac{3 x}{3} = 3 arctan (0.58) + 54 0^{\circ} n - 3 \cdot 4 5^{\circ}

Simplify

\frac{3 x}{3} = 3 arctan (0.58) + 54 0^{\circ} n - 3 \cdot 4 5^{\circ}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

3 arctan (0.58) + 54 0^{\circ} n - 3 \cdot 4 5^{\circ} : 3 arctan (0.58) + 54 0^{\circ} n - 13 5^{\circ}

3 arctan (0.58) + 54 0^{\circ} n - 3 \cdot 4 5^{\circ}

Multiply

3 \cdot 4 5^{\circ} : 13 5^{\circ}

3 \cdot 4 5^{\circ}

Multiply fractions:

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

= 13 5^{\circ}

= 3 arctan (0.58) + 54 0^{\circ} n - 13 5^{\circ}

x = 3 arctan (0.58) + 54 0^{\circ} n - 13 5^{\circ}

x = 3 arctan (0.58) + 54 0^{\circ} n - 13 5^{\circ}

x = 3 arctan (0.58) + 54 0^{\circ} n - 13 5^{\circ}

x = 3 arctan (0.58) + 54 0^{\circ} n - 13 5^{\circ}

tan (4 5^{\circ} + \frac{x}{3}) = - 0.58 : x = - 3 arctan (\frac{29}{5 2}) + 54 0^{\circ} n - 13 5^{\circ}

tan (4 5^{\circ} + \frac{x}{3}) = - 0.58

Apply trig inverse properties

tan (4 5^{\circ} + \frac{x}{3}) = - 0.58

General solutions for

tan (4 5^{\circ} + \frac{x}{3}) = - 0.58

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

4 5^{\circ} + \frac{x}{3} = arctan (- 0.58) + 18 0^{\circ} n

4 5^{\circ} + \frac{x}{3} = arctan (- 0.58) + 18 0^{\circ} n

Solve

4 5^{\circ} + \frac{x}{3} = arctan (- 0.58) + 18 0^{\circ} n : x = - 3 arctan (\frac{29}{5 2}) + 54 0^{\circ} n - 13 5^{\circ}

4 5^{\circ} + \frac{x}{3} = arctan (- 0.58) + 18 0^{\circ} n

Simplify

arctan (- 0.58) + 18 0^{\circ} n : - arctan (\frac{29}{5 2}) + 18 0^{\circ} n

arctan (- 0.58) + 18 0^{\circ} n

arctan (- 0.58) = - arctan (\frac{58}{10})

arctan (- 0.58)

= arctan (- \frac{29}{50})

Use the following property:

arctan (- x) = - arctan (x)

arctan (- \frac{29}{50}) = - arctan (\frac{29}{50})

= - arctan (\frac{29}{50})

= - arctan (\frac{58}{10})

= - arctan (\frac{58}{10}) + 18 0^{\circ} n

\frac{58}{10} = \frac{29}{5 2}

\frac{58}{10}

Factor

58 : 229

Factor

58 = 2 \cdot 29

= 2 \cdot 29

Apply radical rule:

= 229

Factor

10 : 2 \cdot 5

Factor

10 = 2 \cdot 5

= \frac{2 29}{2 \cdot 5}

Cancel

\frac{2 29}{2 \cdot 5} : \frac{29}{2 \cdot 5}

\frac{2 29}{2 \cdot 5}

Apply radical rule:

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

= \frac{2 ^{\frac{1}{2}} 29}{2 \cdot 5}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

= \frac{29}{5 \cdot 2 ^{- \frac{1}{2} + 1}}

Subtract the numbers:

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

= \frac{29}{5 \cdot 2 ^{\frac{1}{2}}}

Apply radical rule:

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

= \frac{29}{5 2}

= \frac{29}{2 \cdot 5}

= - arctan (\frac{29}{5 2}) + 18 0^{\circ} n

4 5^{\circ} + \frac{x}{3} = - arctan (\frac{29}{5 2}) + 18 0^{\circ} n

Move

4 5^{\circ}

to the right side

4 5^{\circ} + \frac{x}{3} = - arctan (\frac{29}{5 2}) + 18 0^{\circ} n

Subtract

4 5^{\circ}

from both sides

4 5^{\circ} + \frac{x}{3} - 4 5^{\circ} = - arctan (\frac{29}{5 2}) + 18 0^{\circ} n - 4 5^{\circ}

Simplify

4 5^{\circ} + \frac{x}{3} - 4 5^{\circ} = - arctan (\frac{29}{5 2}) + 18 0^{\circ} n - 4 5^{\circ}

Simplify

4 5^{\circ} + \frac{x}{3} - 4 5^{\circ} : \frac{x}{3}

4 5^{\circ} + \frac{x}{3} - 4 5^{\circ}

Add similar elements:

4 5^{\circ} - 4 5^{\circ} = 0

= \frac{x}{3}

Simplify

- arctan (\frac{29}{5 2}) + 18 0^{\circ} n - 4 5^{\circ} : - arctan (\frac{29}{5 2}) + 18 0^{\circ} n - 4 5^{\circ}

- arctan (\frac{29}{5 2}) + 18 0^{\circ} n - 4 5^{\circ}

= - arctan (\frac{58}{10}) + 18 0^{\circ} n - 4 5^{\circ}

\frac{58}{10} = \frac{29}{5 2}

\frac{58}{10}

Factor

58 : 229

Factor

58 = 2 \cdot 29

= 2 \cdot 29

Apply radical rule:

= 229

Factor

10 : 2 \cdot 5

Factor

10 = 2 \cdot 5

= \frac{2 29}{2 \cdot 5}

Cancel

\frac{2 29}{2 \cdot 5} : \frac{29}{2 \cdot 5}

\frac{2 29}{2 \cdot 5}

Apply radical rule:

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

= \frac{2 ^{\frac{1}{2}} 29}{2 \cdot 5}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

= \frac{29}{5 \cdot 2 ^{- \frac{1}{2} + 1}}

Subtract the numbers:

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

= \frac{29}{5 \cdot 2 ^{\frac{1}{2}}}

Apply radical rule:

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

= \frac{29}{5 2}

= \frac{29}{2 \cdot 5}

= - arctan (\frac{29}{5 2}) + 18 0^{\circ} n - 4 5^{\circ}

Could not simplify further

= - arctan (\frac{29}{5 2}) + 18 0^{\circ} n - 4 5^{\circ}

\frac{x}{3} = - arctan (\frac{29}{5 2}) + 18 0^{\circ} n - 4 5^{\circ}

\frac{x}{3} = - arctan (\frac{29}{5 2}) + 18 0^{\circ} n - 4 5^{\circ}

\frac{x}{3} = - arctan (\frac{29}{5 2}) + 18 0^{\circ} n - 4 5^{\circ}

Multiply both sides by

3

\frac{x}{3} = - arctan (\frac{29}{5 2}) + 18 0^{\circ} n - 4 5^{\circ}

Multiply both sides by

3

\frac{3 x}{3} = - 3 arctan (\frac{29}{5 2}) + 54 0^{\circ} n - 3 \cdot 4 5^{\circ}

Simplify

\frac{3 x}{3} = - 3 arctan (\frac{29}{5 2}) + 54 0^{\circ} n - 3 \cdot 4 5^{\circ}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

- 3 arctan (\frac{29}{5 2}) + 54 0^{\circ} n - 3 \cdot 4 5^{\circ} : - 3 arctan (\frac{29}{5 2}) + 54 0^{\circ} n - 13 5^{\circ}

- 3 arctan (\frac{29}{5 2}) + 54 0^{\circ} n - 3 \cdot 4 5^{\circ}

Simplify

arctan (\frac{29}{5 2}) : arctan (\frac{58}{10})

arctan (\frac{29}{5 2})

= arctan (\frac{58}{10})

= - 3 arctan (\frac{58}{10}) + 54 0^{\circ} n - 3 \cdot 4 5^{\circ}

Multiply

3 \cdot 4 5^{\circ} : 13 5^{\circ}

3 \cdot 4 5^{\circ}

Multiply fractions:

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

= 13 5^{\circ}

= - 3 arctan (\frac{58}{10}) + 54 0^{\circ} n - 13 5^{\circ}

\frac{58}{10} = \frac{29}{5 2}

\frac{58}{10}

Factor

58 : 229

Factor

58 = 2 \cdot 29

= 2 \cdot 29

Apply radical rule:

= 229

Factor

10 : 2 \cdot 5

Factor

10 = 2 \cdot 5

= \frac{2 29}{2 \cdot 5}

Cancel

\frac{2 29}{2 \cdot 5} : \frac{29}{2 \cdot 5}

\frac{2 29}{2 \cdot 5}

Apply radical rule:

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

= \frac{2 ^{\frac{1}{2}} 29}{2 \cdot 5}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

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

= \frac{29}{5 \cdot 2 ^{- \frac{1}{2} + 1}}

Subtract the numbers:

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

= \frac{29}{5 \cdot 2 ^{\frac{1}{2}}}

Apply radical rule:

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

= \frac{29}{5 2}

= \frac{29}{2 \cdot 5}

= - 3 arctan (\frac{29}{5 2}) + 54 0^{\circ} n - 13 5^{\circ}

x = - 3 arctan (\frac{29}{5 2}) + 54 0^{\circ} n - 13 5^{\circ}

x = - 3 arctan (\frac{29}{5 2}) + 54 0^{\circ} n - 13 5^{\circ}

x = - 3 arctan (\frac{29}{5 2}) + 54 0^{\circ} n - 13 5^{\circ}

x = - 3 arctan (\frac{29}{5 2}) + 54 0^{\circ} n - 13 5^{\circ}

Combine all the solutions

x = 3 arctan (0.58) + 54 0^{\circ} n - 13 5^{\circ}, x = - 3 arctan (\frac{29}{5 2}) + 54 0^{\circ} n - 13 5^{\circ}

Show solutions in decimal form

x = 3 \cdot 0.65086 \dots + 54 0^{\circ} n - 13 5^{\circ}, x = - 3 \cdot 0.65086 \dots + 54 0^{\circ} n - 13 5^{\circ}

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Frequently Asked Questions (FAQ)

What is the general solution for tan^2(45+x/3)=0.58 ?
The general solution for tan^2(45+x/3)=0.58 is x=3*0.65086…+540n-135,x=-3*0.65086…+540n-135