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Popular >

tan^2(x)+tan(x)-12=0,0<= x<= 2pi

Solution

tan^{2} (x) + tan (x) - 12 = 0, 0 \leq x \leq 2 π

Solution

x = 1.24904 \dots, x = 1.24904 \dots + π, x = - 1.32581 \dots + π, x = - 1.32581 \dots + 2 π

+1

Degrees

x = 71.56505 \dots^{\circ}, x = 251.56505 \dots^{\circ}, x = 104.03624 \dots^{\circ}, x = 284.03624 \dots^{\circ}

Solution steps

tan^{2} (x) + tan (x) - 12 = 0, 0 \leq x \leq 2 π

Solve by substitution

tan^{2} (x) + tan (x) - 12 = 0

Let:

tan (x) = u

u^{2} + u - 12 = 0

u^{2} + u - 12 = 0 : u = 3, u = - 4

u^{2} + u - 12 = 0

Solve with the quadratic formula

u^{2} + u - 12 = 0

Quadratic Equation Formula:

For

a = 1, b = 1, c = - 12

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 1 \cdot ( - 12 )}{2 \cdot 1}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 1 \cdot ( - 12 )}{2 \cdot 1}

1^{2} - 4 \cdot 1 \cdot (- 12) = 7

1^{2} - 4 \cdot 1 \cdot (- 12)

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 1 \cdot (- 12)

Apply rule

- (- a) = a

= 1 + 4 \cdot 1 \cdot 12

Multiply the numbers:

4 \cdot 1 \cdot 12 = 48

= 1 + 48

Add the numbers:

1 + 48 = 49

= 49

Factor the number:

49 = 7^{2}

= 7^{2}

Apply radical rule:

n a^{n} = a

7^{2} = 7

= 7

u_{1, 2} = \frac{- 1 \pm 7}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- 1 + 7}{2 \cdot 1}, u_{2} = \frac{- 1 - 7}{2 \cdot 1}

u = \frac{- 1 + 7}{2 \cdot 1} : 3

\frac{- 1 + 7}{2 \cdot 1}

Add/Subtract the numbers:

- 1 + 7 = 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{- 1 - 7}{2 \cdot 1} : - 4

\frac{- 1 - 7}{2 \cdot 1}

Subtract the numbers:

- 1 - 7 = - 8

= \frac{- 8}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 8}{2}

Apply the fraction rule:

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

= - \frac{8}{2}

Divide the numbers:

\frac{8}{2} = 4

= - 4

The solutions to the quadratic equation are:

u = 3, u = - 4

Substitute back

u = tan (x)

tan (x) = 3, tan (x) = - 4

tan (x) = 3, tan (x) = - 4

tan (x) = 3, 0 \leq x \leq 2 π : x = arctan (3), x = arctan (3) + π

tan (x) = 3, 0 \leq x \leq 2 π

Apply trig inverse properties

tan (x) = 3

General solutions for

tan (x) = 3

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

x = arctan (3) + πn

x = arctan (3) + πn

Solutions for the range

0 \leq x \leq 2 π

x = arctan (3), x = arctan (3) + π

tan (x) = - 4, 0 \leq x \leq 2 π : x = - arctan (4) + π, x = - arctan (4) + 2 π

tan (x) = - 4, 0 \leq x \leq 2 π

Apply trig inverse properties

tan (x) = - 4

General solutions for

tan (x) = - 4

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

x = arctan (- 4) + πn

x = arctan (- 4) + πn

Solutions for the range

0 \leq x \leq 2 π

x = - arctan (4) + π, x = - arctan (4) + 2 π

Combine all the solutions

x = arctan (3), x = arctan (3) + π, x = - arctan (4) + π, x = - arctan (4) + 2 π

Show solutions in decimal form

x = 1.24904 \dots, x = 1.24904 \dots + π, x = - 1.32581 \dots + π, x = - 1.32581 \dots + 2 π

Graph

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