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

The general solution for -3tan^2(θ)-4tan(θ)=-4 is θ=-1.10714…+pin,θ=0.58800…+pin

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Popular >

-3tan^2(θ)-4tan(θ)=-4

Solution

- 3 tan^{2} (θ) - 4 tan (θ) = - 4

Solution

θ = - 1.10714 \dots + πn, θ = 0.58800 \dots + πn

Degrees

θ = - 63.43494 \dots^{\circ} + 18 0^{\circ} n, θ = 33.69006 \dots^{\circ} + 18 0^{\circ} n

Solution steps

- 3 tan^{2} (θ) - 4 tan (θ) = - 4

Solve by substitution

- 3 tan^{2} (θ) - 4 tan (θ) = - 4

Let:

tan (θ) = u

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

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

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

Move

4

to the left side

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

Add

4

to both sides

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

Simplify

- 3 u^{2} - 4 u + 4 = 0

- 3 u^{2} - 4 u + 4 = 0

Solve with the quadratic formula

- 3 u^{2} - 4 u + 4 = 0

Quadratic Equation Formula:

For

a = - 3, b = - 4, c = 4

u_{1, 2} = \frac{- ( - 4 ) \pm ( - 4 ) ^{2} - 4 ( - 3 ) \cdot 4}{2 ( - 3 )}

u_{1, 2} = \frac{- ( - 4 ) \pm ( - 4 ) ^{2} - 4 ( - 3 ) \cdot 4}{2 ( - 3 )}

(- 4)^{2} - 4 (- 3) \cdot 4 = 8

(- 4)^{2} - 4 (- 3) \cdot 4

Apply rule

- (- a) = a

= (- 4)^{2} + 4 \cdot 3 \cdot 4

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 4)^{2} = 4^{2}

= 4^{2} + 4 \cdot 3 \cdot 4

Multiply the numbers:

4 \cdot 3 \cdot 4 = 48

= 4^{2} + 48

4^{2} = 16

= 16 + 48

Add the numbers:

16 + 48 = 64

= 64

Factor the number:

64 = 8^{2}

= 8^{2}

Apply radical rule:

n a^{n} = a

8^{2} = 8

= 8

u_{1, 2} = \frac{- ( - 4 ) \pm 8}{2 ( - 3 )}

Separate the solutions

u_{1} = \frac{- ( - 4 ) + 8}{2 ( - 3 )}, u_{2} = \frac{- ( - 4 ) - 8}{2 ( - 3 )}

u = \frac{- ( - 4 ) + 8}{2 ( - 3 )} : - 2

\frac{- ( - 4 ) + 8}{2 ( - 3 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{4 + 8}{- 2 \cdot 3}

Add the numbers:

4 + 8 = 12

= \frac{12}{- 2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{12}{- 6}

Apply the fraction rule:

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

= - \frac{12}{6}

Divide the numbers:

\frac{12}{6} = 2

= - 2

u = \frac{- ( - 4 ) - 8}{2 ( - 3 )} : \frac{2}{3}

\frac{- ( - 4 ) - 8}{2 ( - 3 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{4 - 8}{- 2 \cdot 3}

Subtract the numbers:

4 - 8 = - 4

= \frac{- 4}{- 2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 4}{- 6}

Apply the fraction rule:

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

= \frac{4}{6}

Cancel the common factor:

2

= \frac{2}{3}

The solutions to the quadratic equation are:

u = - 2, u = \frac{2}{3}

Substitute back

u = tan (θ)

tan (θ) = - 2, tan (θ) = \frac{2}{3}

tan (θ) = - 2, tan (θ) = \frac{2}{3}

tan (θ) = - 2 : θ = arctan (- 2) + πn

tan (θ) = - 2

Apply trig inverse properties

tan (θ) = - 2

General solutions for

tan (θ) = - 2

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

θ = arctan (- 2) + πn

θ = arctan (- 2) + πn

tan (θ) = \frac{2}{3} : θ = arctan (\frac{2}{3}) + πn

tan (θ) = \frac{2}{3}

Apply trig inverse properties

tan (θ) = \frac{2}{3}

General solutions for

tan (θ) = \frac{2}{3}

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

θ = arctan (\frac{2}{3}) + πn

θ = arctan (\frac{2}{3}) + πn

Combine all the solutions

θ = arctan (- 2) + πn, θ = arctan (\frac{2}{3}) + πn

Show solutions in decimal form

θ = - 1.10714 \dots + πn, θ = 0.58800 \dots + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for -3tan^2(θ)-4tan(θ)=-4 ?
The general solution for -3tan^2(θ)-4tan(θ)=-4 is θ=-1.10714…+pin,θ=0.58800…+pin