What is the general solution for 4+2tan^2(θ)=5tan^2(θ)+3 ?

The general solution for 4+2tan^2(θ)=5tan^2(θ)+3 is θ=0.52359…+pin,θ=-0.52359…+pin

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Popular >

4+2tan^2(θ)=5tan^2(θ)+3

Solution

4 + 2 tan^{2} (θ) = 5 tan^{2} (θ) + 3

Solution

θ = 0.52359 \dots + πn, θ = - 0.52359 \dots + πn

Degrees

θ = 3 0^{\circ} + 18 0^{\circ} n, θ = - 3 0^{\circ} + 18 0^{\circ} n

Solution steps

4 + 2 tan^{2} (θ) = 5 tan^{2} (θ) + 3

Solve by substitution

4 + 2 tan^{2} (θ) = 5 tan^{2} (θ) + 3

Let:

tan (θ) = u

4 + 2 u^{2} = 5 u^{2} + 3

4 + 2 u^{2} = 5 u^{2} + 3 : u = \frac{1}{3}, u = - \frac{1}{3}

4 + 2 u^{2} = 5 u^{2} + 3

Move

4

to the right side

4 + 2 u^{2} = 5 u^{2} + 3

Subtract

4

from both sides

4 + 2 u^{2} - 4 = 5 u^{2} + 3 - 4

Simplify

2 u^{2} = 5 u^{2} - 1

2 u^{2} = 5 u^{2} - 1

Move

5 u^{2}

to the left side

2 u^{2} = 5 u^{2} - 1

Subtract

5 u^{2}

from both sides

2 u^{2} - 5 u^{2} = 5 u^{2} - 1 - 5 u^{2}

Simplify

- 3 u^{2} = - 1

- 3 u^{2} = - 1

Divide both sides by

- 3

- 3 u^{2} = - 1

Divide both sides by

- 3

\frac{- 3 u ^{2}}{- 3} = \frac{- 1}{- 3}

Simplify

u^{2} = \frac{1}{3}

u^{2} = \frac{1}{3}

For

x^{2} = f (a)

the solutions are

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

u = \frac{1}{3}, u = - \frac{1}{3}

Substitute back

u = tan (θ)

tan (θ) = \frac{1}{3}, tan (θ) = - \frac{1}{3}

tan (θ) = \frac{1}{3}, tan (θ) = - \frac{1}{3}

tan (θ) = \frac{1}{3} : θ = arctan (\frac{1}{3}) + πn

tan (θ) = \frac{1}{3}

Apply trig inverse properties

tan (θ) = \frac{1}{3}

General solutions for

tan (θ) = \frac{1}{3}

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

θ = arctan (\frac{1}{3}) + πn

θ = arctan (\frac{1}{3}) + πn

tan (θ) = - \frac{1}{3} : θ = arctan (- \frac{1}{3}) + πn

tan (θ) = - \frac{1}{3}

Apply trig inverse properties

tan (θ) = - \frac{1}{3}

General solutions for

tan (θ) = - \frac{1}{3}

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

θ = arctan (- \frac{1}{3}) + πn

θ = arctan (- \frac{1}{3}) + πn

Combine all the solutions

θ = arctan (\frac{1}{3}) + πn, θ = arctan (- \frac{1}{3}) + πn

Show solutions in decimal form

θ = 0.52359 \dots + πn, θ = - 0.52359 \dots + πn

Popular Examples

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sin (x) = sin (\frac{π}{5})

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x, sin (2 x) = cos (2 x)

cos(x)= 16/15

cos (x) = \frac{16}{15}

2sec^2(x)-3tan(x)=2

2 sec^{2} (x) - 3 tan (x) = 2

0=2(1+cos(θ))

0 = 2 (1 + cos (θ))

Frequently Asked Questions (FAQ)

What is the general solution for 4+2tan^2(θ)=5tan^2(θ)+3 ?
The general solution for 4+2tan^2(θ)=5tan^2(θ)+3 is θ=0.52359…+pin,θ=-0.52359…+pin