What is the general solution for tan^2(θ)-3tan(θ)=0 ?

The general solution for tan^2(θ)-3tan(θ)=0 is θ=1.24904…+pin,θ=pin

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tan^2(θ)-3tan(θ)=0

Solution

tan^{2} (θ) - 3 tan (θ) = 0

Solution

θ = 1.24904 \dots + πn, θ = πn

Degrees

θ = 71.56505 \dots^{\circ} + 18 0^{\circ} n, θ = 0^{\circ} + 18 0^{\circ} n

Solution steps

tan^{2} (θ) - 3 tan (θ) = 0

Solve by substitution

tan^{2} (θ) - 3 tan (θ) = 0

Let:

tan (θ) = u

u^{2} - 3 u = 0

u^{2} - 3 u = 0 : u = 3, u = 0

u^{2} - 3 u = 0

Solve with the quadratic formula

u^{2} - 3 u = 0

Quadratic Equation Formula:

For

a = 1, b = - 3, c = 0

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 1 \cdot 0}{2 \cdot 1}

(- 3)^{2} - 4 \cdot 1 \cdot 0 = 3

(- 3)^{2} - 4 \cdot 1 \cdot 0

Apply exponent rule:

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

n

is even

(- 3)^{2} = 3^{2}

= 3^{2} - 4 \cdot 1 \cdot 0

Apply rule

0 \cdot a = 0

= 3^{2} - 0

3^{2} - 0 = 3^{2}

= 3^{2}

Apply radical rule:

assuming

a \geq 0

= 3

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

Separate the solutions

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

u = \frac{- ( - 3 ) + 3}{2 \cdot 1} : 3

\frac{- ( - 3 ) + 3}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{3 + 3}{2 \cdot 1}

Add the numbers:

3 + 3 = 6

= \frac{6}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{6}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3

u = \frac{- ( - 3 ) - 3}{2 \cdot 1} : 0

\frac{- ( - 3 ) - 3}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{3 - 3}{2 \cdot 1}

Subtract the numbers:

3 - 3 = 0

= \frac{0}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{0}{2}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

The solutions to the quadratic equation are:

u = 3, u = 0

Substitute back

u = tan (θ)

tan (θ) = 3, tan (θ) = 0

tan (θ) = 3, tan (θ) = 0

tan (θ) = 3 : θ = arctan (3) + πn

tan (θ) = 3

Apply trig inverse properties

tan (θ) = 3

General solutions for

tan (θ) = 3

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

θ = arctan (3) + πn

θ = arctan (3) + πn

tan (θ) = 0 : θ = πn

tan (θ) = 0

General solutions for

tan (θ) = 0

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

θ = 0 + πn

θ = 0 + πn

Solve

θ = 0 + πn : θ = πn

θ = 0 + πn

0 + πn = πn

θ = πn

θ = πn

Combine all the solutions

θ = arctan (3) + πn, θ = πn

Show solutions in decimal form

θ = 1.24904 \dots + πn, θ = πn

Popular Examples

sin^2(x)=2cos(x)+2

sin^{2} (x) = 2 cos (x) + 2

2cos^2(x)+3cos(x)=0

2 cos^{2} (x) + 3 cos (x) = 0

solvefor y,sin(y)=x solve for

y, sin (y) = x

sin(2x)+sqrt(3)cos(x)=0

sin (2 x) + 3 cos (x) = 0

tan(θ)=cot(θ)

tan (θ) = cot (θ)

Frequently Asked Questions (FAQ)

What is the general solution for tan^2(θ)-3tan(θ)=0 ?
The general solution for tan^2(θ)-3tan(θ)=0 is θ=1.24904…+pin,θ=pin