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

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

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189.4=100tan^2(45+θ/2)

Solution

189.4 = 100 tan^{2} (4 5^{\circ} + \frac{θ}{2})

Solution

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

Radians

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

Solution steps

189.4 = 100 tan^{2} (4 5^{\circ} + \frac{θ}{2})

Switch sides

100 tan^{2} (4 5^{\circ} + \frac{θ}{2}) = 189.4

Solve by substitution

100 tan^{2} (4 5^{\circ} + \frac{θ}{2}) = 189.4

Let:

tan (4 5^{\circ} + \frac{θ}{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{θ}{2})

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

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

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

tan (4 5^{\circ} + \frac{θ}{2}) = 1.894

Apply trig inverse properties

tan (4 5^{\circ} + \frac{θ}{2}) = 1.894

General solutions for

tan (4 5^{\circ} + \frac{θ}{2}) = 1.894

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

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

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

Solve

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

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

Move

4 5^{\circ}

to the right side

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

Subtract

4 5^{\circ}

from both sides

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

Simplify

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

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

Multiply both sides by

2

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

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

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}

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

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

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

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

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

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

Apply trig inverse properties

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

General solutions for

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

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

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

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

Solve

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

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

Move

4 5^{\circ}

to the right side

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

Subtract

4 5^{\circ}

from both sides

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

Simplify

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

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

Multiply both sides by

2

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

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 θ}{2} : θ

\frac{2 θ}{2}

Divide the numbers:

\frac{2}{2} = 1

= θ

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}

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

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

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

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

Combine all the solutions

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

Show solutions in decimal form

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

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Frequently Asked Questions (FAQ)

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