What is the general solution for -4tan^2(θ)=-1 ?

The general solution for -4tan^2(θ)=-1 is θ=0.46364…+pin,θ=-0.46364…+pin

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Popular >

-4tan^2(θ)=-1

Solution

- 4 tan^{2} (θ) = - 1

Solution

θ = 0.46364 \dots + πn, θ = - 0.46364 \dots + πn

Degrees

θ = 26.56505 \dots^{\circ} + 18 0^{\circ} n, θ = - 26.56505 \dots^{\circ} + 18 0^{\circ} n

Solution steps

- 4 tan^{2} (θ) = - 1

Solve by substitution

- 4 tan^{2} (θ) = - 1

Let:

tan (θ) = u

- 4 u^{2} = - 1

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

- 4 u^{2} = - 1

Divide both sides by

- 4

- 4 u^{2} = - 1

Divide both sides by

- 4

\frac{- 4 u ^{2}}{- 4} = \frac{- 1}{- 4}

Simplify

u^{2} = \frac{1}{4}

u^{2} = \frac{1}{4}

For

x^{2} = f (a)

the solutions are

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

u = \frac{1}{4}, u = - \frac{1}{4}

\frac{1}{4} = \frac{1}{2}

\frac{1}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{1}{2}

Apply rule

1 = 1

= \frac{1}{2}

- \frac{1}{4} = - \frac{1}{2}

- \frac{1}{4}

Simplify

\frac{1}{4} : \frac{1}{2}

\frac{1}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{1}{2}

= - \frac{1}{2}

Apply rule

1 = 1

= - \frac{1}{2}

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

Substitute back

u = tan (θ)

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

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

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

tan (θ) = \frac{1}{2}

Apply trig inverse properties

tan (θ) = \frac{1}{2}

General solutions for

tan (θ) = \frac{1}{2}

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

θ = arctan (\frac{1}{2}) + πn

θ = arctan (\frac{1}{2}) + πn

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

tan (θ) = - \frac{1}{2}

Apply trig inverse properties

tan (θ) = - \frac{1}{2}

General solutions for

tan (θ) = - \frac{1}{2}

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

θ = arctan (- \frac{1}{2}) + πn

θ = arctan (- \frac{1}{2}) + πn

Combine all the solutions

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

Show solutions in decimal form

θ = 0.46364 \dots + πn, θ = - 0.46364 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for -4tan^2(θ)=-1 ?
The general solution for -4tan^2(θ)=-1 is θ=0.46364…+pin,θ=-0.46364…+pin