What is the general solution for 189.4=100tan^2(45-x/2) ?

The general solution for 189.4=100tan^2(45-x/2) is x=-360n+90-2*0.94242…,x=-360n+90+2*0.94242…

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189.4=100tan^2(45-x/2)

Solution

189.4 = 100 tan^{2} (4 5^{\circ} - \frac{x}{2})

Solution

x = - 36 0^{\circ} n + 9 0^{\circ} - 2 \cdot 0.94242 \dots, x = - 36 0^{\circ} n + 9 0^{\circ} + 2 \cdot 0.94242 \dots

Radians

x = \frac{π}{2} - 2 \cdot 0.94242 \dots - 2 πn, x = \frac{π}{2} + 2 \cdot 0.94242 \dots - 2 πn

Solution steps

189.4 = 100 tan^{2} (4 5^{\circ} - \frac{x}{2})

Switch sides

100 tan^{2} (4 5^{\circ} - \frac{x}{2}) = 189.4

Solve by substitution

100 tan^{2} (4 5^{\circ} - \frac{x}{2}) = 189.4

Let:

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

100 u^{2} = 189.4

100 u^{2} = 189.4 : u = 1.894, u = - 1.894

100 u^{2} = 189.4

Divide both sides by

100

100 u^{2} = 189.4

Divide both sides by

100

\frac{100 u ^{2}}{100} = \frac{189.4}{100}

Simplify

u^{2} = 1.894

u^{2} = 1.894

For

x^{2} = f (a)

the solutions are

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

u = 1.894, u = - 1.894

Substitute back

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

tan (4 5^{\circ} - \frac{x}{2}) = 1.894, tan (4 5^{\circ} - \frac{x}{2}) = - 1.894

tan (4 5^{\circ} - \frac{x}{2}) = 1.894, tan (4 5^{\circ} - \frac{x}{2}) = - 1.894

tan (4 5^{\circ} - \frac{x}{2}) = 1.894 : x = - 36 0^{\circ} n + 9 0^{\circ} - 2 arctan (1.894)

tan (4 5^{\circ} - \frac{x}{2}) = 1.894

Apply trig inverse properties

tan (4 5^{\circ} - \frac{x}{2}) = 1.894

General solutions for

tan (4 5^{\circ} - \frac{x}{2}) = 1.894

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

4 5^{\circ} - \frac{x}{2} = arctan (1.894) + 18 0^{\circ} n

4 5^{\circ} - \frac{x}{2} = arctan (1.894) + 18 0^{\circ} n

Solve

4 5^{\circ} - \frac{x}{2} = arctan (1.894) + 18 0^{\circ} n : x = - 36 0^{\circ} n + 9 0^{\circ} - 2 arctan (1.894)

4 5^{\circ} - \frac{x}{2} = arctan (1.894) + 18 0^{\circ} n

Move

4 5^{\circ}

to the right side

4 5^{\circ} - \frac{x}{2} = arctan (1.894) + 18 0^{\circ} n

Subtract

4 5^{\circ}

from both sides

4 5^{\circ} - \frac{x}{2} - 4 5^{\circ} = arctan (1.894) + 18 0^{\circ} n - 4 5^{\circ}

Simplify

- \frac{x}{2} = arctan (1.894) + 18 0^{\circ} n - 4 5^{\circ}

- \frac{x}{2} = arctan (1.894) + 18 0^{\circ} n - 4 5^{\circ}

Multiply both sides by

2

- \frac{x}{2} = arctan (1.894) + 18 0^{\circ} n - 4 5^{\circ}

Multiply both sides by

2

2 (- \frac{x}{2}) = 2 arctan (1.894) + 36 0^{\circ} n - 2 \cdot 4 5^{\circ}

Simplify

2 (- \frac{x}{2}) = 2 arctan (1.894) + 36 0^{\circ} n - 2 \cdot 4 5^{\circ}

Simplify

2 (- \frac{x}{2}) : - x

2 (- \frac{x}{2})

Remove parentheses:

(- a) = - a

= - 2 \cdot \frac{x}{2}

Multiply fractions:

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

= - \frac{x \cdot 2}{2}

Cancel the common factor:

2

= - x

Simplify

2 arctan (1.894) + 36 0^{\circ} n - 2 \cdot 4 5^{\circ} : 2 arctan (1.894) + 36 0^{\circ} n - 9 0^{\circ}

2 arctan (1.894) + 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 (1.894) + 36 0^{\circ} n - 9 0^{\circ}

- x = 2 arctan (1.894) + 36 0^{\circ} n - 9 0^{\circ}

- x = 2 arctan (1.894) + 36 0^{\circ} n - 9 0^{\circ}

- x = 2 arctan (1.894) + 36 0^{\circ} n - 9 0^{\circ}

Divide both sides by

- 1

- x = 2 arctan (1.894) + 36 0^{\circ} n - 9 0^{\circ}

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{2 arctan ( 1.894 )}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{9 0 ^{\circ}}{- 1}

Simplify

\frac{- x}{- 1} = \frac{2 arctan ( 1.894 )}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{9 0 ^{\circ}}{- 1}

Simplify

\frac{- x}{- 1} : x

\frac{- x}{- 1}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{x}{1}

Apply rule

\frac{a}{1} = a

= x

Simplify

\frac{2 arctan ( 1.894 )}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{9 0 ^{\circ}}{- 1} : - 36 0^{\circ} n + 9 0^{\circ} - 2 arctan (1.894)

\frac{2 arctan ( 1.894 )}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{9 0 ^{\circ}}{- 1}

Group like terms

= \frac{36 0 ^{\circ} n}{- 1} - \frac{9 0 ^{\circ}}{- 1} + \frac{2 arctan ( 1.894 )}{- 1}

\frac{36 0 ^{\circ} n}{- 1} = - 36 0^{\circ} n

\frac{36 0 ^{\circ} n}{- 1}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{36 0 ^{\circ} n}{1}

Apply rule

\frac{a}{1} = a

= - 36 0^{\circ} n

= - 36 0^{\circ} n - \frac{9 0 ^{\circ}}{- 1} + \frac{2 arctan ( 1.894 )}{- 1}

\frac{9 0 ^{\circ}}{- 1} = - 9 0^{\circ}

\frac{9 0 ^{\circ}}{- 1}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{9 0 ^{\circ}}{1}

Apply the fraction rule:

\frac{a}{1} = a

\frac{9 0 ^{\circ}}{1} = 9 0^{\circ}

= - 9 0^{\circ}

\frac{2 arctan ( 1.894 )}{- 1} = - 2 arctan (1.894)

\frac{2 arctan ( 1.894 )}{- 1}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{2 arctan ( 1.894 )}{1}

Apply rule

\frac{a}{1} = a

= - 2 arctan (1.894)

= - 36 0^{\circ} n - (- 9 0^{\circ}) - 2 arctan (1.894)

Apply rule

- (- a) = a

= - 36 0^{\circ} n + 9 0^{\circ} - 2 arctan (1.894)

x = - 36 0^{\circ} n + 9 0^{\circ} - 2 arctan (1.894)

x = - 36 0^{\circ} n + 9 0^{\circ} - 2 arctan (1.894)

x = - 36 0^{\circ} n + 9 0^{\circ} - 2 arctan (1.894)

x = - 36 0^{\circ} n + 9 0^{\circ} - 2 arctan (1.894)

tan (4 5^{\circ} - \frac{x}{2}) = - 1.894 : x = - 36 0^{\circ} n + 9 0^{\circ} + 2 arctan (1.894)

tan (4 5^{\circ} - \frac{x}{2}) = - 1.894

Apply trig inverse properties

tan (4 5^{\circ} - \frac{x}{2}) = - 1.894

General solutions for

tan (4 5^{\circ} - \frac{x}{2}) = - 1.894

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

4 5^{\circ} - \frac{x}{2} = arctan (- 1.894) + 18 0^{\circ} n

4 5^{\circ} - \frac{x}{2} = arctan (- 1.894) + 18 0^{\circ} n

Solve

4 5^{\circ} - \frac{x}{2} = arctan (- 1.894) + 18 0^{\circ} n : x = - 36 0^{\circ} n + 9 0^{\circ} + 2 arctan (1.894)

4 5^{\circ} - \frac{x}{2} = arctan (- 1.894) + 18 0^{\circ} n

Simplify

arctan (- 1.894) + 18 0^{\circ} n : - arctan (1.894) + 18 0^{\circ} n

arctan (- 1.894) + 18 0^{\circ} n

Use the following property:

arctan (- x) = - arctan (x)

arctan (- 1.894) = - arctan (1.894)

= - arctan (1.894) + 18 0^{\circ} n

4 5^{\circ} - \frac{x}{2} = - arctan (1.894) + 18 0^{\circ} n

Move

4 5^{\circ}

to the right side

4 5^{\circ} - \frac{x}{2} = - arctan (1.894) + 18 0^{\circ} n

Subtract

4 5^{\circ}

from both sides

4 5^{\circ} - \frac{x}{2} - 4 5^{\circ} = - arctan (1.894) + 18 0^{\circ} n - 4 5^{\circ}

Simplify

- \frac{x}{2} = - arctan (1.894) + 18 0^{\circ} n - 4 5^{\circ}

- \frac{x}{2} = - arctan (1.894) + 18 0^{\circ} n - 4 5^{\circ}

Multiply both sides by

2

- \frac{x}{2} = - arctan (1.894) + 18 0^{\circ} n - 4 5^{\circ}

Multiply both sides by

2

2 (- \frac{x}{2}) = - 2 arctan (1.894) + 36 0^{\circ} n - 2 \cdot 4 5^{\circ}

Simplify

2 (- \frac{x}{2}) = - 2 arctan (1.894) + 36 0^{\circ} n - 2 \cdot 4 5^{\circ}

Simplify

2 (- \frac{x}{2}) : - x

2 (- \frac{x}{2})

Remove parentheses:

(- a) = - a

= - 2 \cdot \frac{x}{2}

Multiply fractions:

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

= - \frac{x \cdot 2}{2}

Cancel the common factor:

2

= - x

Simplify

- 2 arctan (1.894) + 36 0^{\circ} n - 2 \cdot 4 5^{\circ} : - 2 arctan (1.894) + 36 0^{\circ} n - 9 0^{\circ}

- 2 arctan (1.894) + 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 (1.894) + 36 0^{\circ} n - 9 0^{\circ}

- x = - 2 arctan (1.894) + 36 0^{\circ} n - 9 0^{\circ}

- x = - 2 arctan (1.894) + 36 0^{\circ} n - 9 0^{\circ}

- x = - 2 arctan (1.894) + 36 0^{\circ} n - 9 0^{\circ}

Divide both sides by

- 1

- x = - 2 arctan (1.894) + 36 0^{\circ} n - 9 0^{\circ}

Divide both sides by

- 1

\frac{- x}{- 1} = - \frac{2 arctan ( 1.894 )}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{9 0 ^{\circ}}{- 1}

Simplify

\frac{- x}{- 1} = - \frac{2 arctan ( 1.894 )}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{9 0 ^{\circ}}{- 1}

Simplify

\frac{- x}{- 1} : x

\frac{- x}{- 1}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{x}{1}

Apply rule

\frac{a}{1} = a

= x

Simplify

- \frac{2 arctan ( 1.894 )}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{9 0 ^{\circ}}{- 1} : - 36 0^{\circ} n + 9 0^{\circ} + 2 arctan (1.894)

- \frac{2 arctan ( 1.894 )}{- 1} + \frac{36 0 ^{\circ} n}{- 1} - \frac{9 0 ^{\circ}}{- 1}

Group like terms

= \frac{36 0 ^{\circ} n}{- 1} - \frac{9 0 ^{\circ}}{- 1} - \frac{2 arctan ( 1.894 )}{- 1}

\frac{36 0 ^{\circ} n}{- 1} = - 36 0^{\circ} n

\frac{36 0 ^{\circ} n}{- 1}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{36 0 ^{\circ} n}{1}

Apply rule

\frac{a}{1} = a

= - 36 0^{\circ} n

= - 36 0^{\circ} n - \frac{9 0 ^{\circ}}{- 1} - \frac{2 arctan ( 1.894 )}{- 1}

\frac{9 0 ^{\circ}}{- 1} = - 9 0^{\circ}

\frac{9 0 ^{\circ}}{- 1}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{9 0 ^{\circ}}{1}

Apply the fraction rule:

\frac{a}{1} = a

\frac{9 0 ^{\circ}}{1} = 9 0^{\circ}

= - 9 0^{\circ}

\frac{2 arctan ( 1.894 )}{- 1} = - 2 arctan (1.894)

\frac{2 arctan ( 1.894 )}{- 1}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{2 arctan ( 1.894 )}{1}

Apply rule

\frac{a}{1} = a

= - 2 arctan (1.894)

= - 36 0^{\circ} n - (- 9 0^{\circ}) - (- 2 arctan (1.894))

Apply rule

- (- a) = a

= - 36 0^{\circ} n + 9 0^{\circ} + 2 arctan (1.894)

x = - 36 0^{\circ} n + 9 0^{\circ} + 2 arctan (1.894)

x = - 36 0^{\circ} n + 9 0^{\circ} + 2 arctan (1.894)

x = - 36 0^{\circ} n + 9 0^{\circ} + 2 arctan (1.894)

x = - 36 0^{\circ} n + 9 0^{\circ} + 2 arctan (1.894)

Combine all the solutions

x = - 36 0^{\circ} n + 9 0^{\circ} - 2 arctan (1.894), x = - 36 0^{\circ} n + 9 0^{\circ} + 2 arctan (1.894)

Show solutions in decimal form

x = - 36 0^{\circ} n + 9 0^{\circ} - 2 \cdot 0.94242 \dots, x = - 36 0^{\circ} n + 9 0^{\circ} + 2 \cdot 0.94242 \dots

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Frequently Asked Questions (FAQ)

What is the general solution for 189.4=100tan^2(45-x/2) ?
The general solution for 189.4=100tan^2(45-x/2) is x=-360n+90-2*0.94242…,x=-360n+90+2*0.94242…