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Popular >

tan^4(x)-13tan^2(x)+36=0

Solution

tan^{4} (x) - 13 tan^{2} (x) + 36 = 0

Solution

x = 1.24904 \dots + πn, x = - 1.24904 \dots + πn, x = 1.10714 \dots + πn, x = - 1.10714 \dots + πn

Degrees

x = 71.56505 \dots^{\circ} + 18 0^{\circ} n, x = - 71.56505 \dots^{\circ} + 18 0^{\circ} n, x = 63.43494 \dots^{\circ} + 18 0^{\circ} n, x = - 63.43494 \dots^{\circ} + 18 0^{\circ} n

Solution steps

tan^{4} (x) - 13 tan^{2} (x) + 36 = 0

Solve by substitution

tan^{4} (x) - 13 tan^{2} (x) + 36 = 0

Let:

tan (x) = u

u^{4} - 13 u^{2} + 36 = 0

u^{4} - 13 u^{2} + 36 = 0 : u = 3, u = - 3, u = 2, u = - 2

u^{4} - 13 u^{2} + 36 = 0

Rewrite the equation with

v = u^{2}

and

v^{2} = u^{4}

v^{2} - 13 v + 36 = 0

Solve

v^{2} - 13 v + 36 = 0 : v = 9, v = 4

v^{2} - 13 v + 36 = 0

Solve with the quadratic formula

v^{2} - 13 v + 36 = 0

Quadratic Equation Formula:

For

a = 1, b = - 13, c = 36

v_{1, 2} = \frac{- ( - 13 ) \pm ( - 13 ) ^{2} - 4 \cdot 1 \cdot 36}{2 \cdot 1}

v_{1, 2} = \frac{- ( - 13 ) \pm ( - 13 ) ^{2} - 4 \cdot 1 \cdot 36}{2 \cdot 1}

(- 13)^{2} - 4 \cdot 1 \cdot 36 = 5

(- 13)^{2} - 4 \cdot 1 \cdot 36

Apply exponent rule:

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

n

is even

(- 13)^{2} = 1 3^{2}

= 1 3^{2} - 4 \cdot 1 \cdot 36

Multiply the numbers:

4 \cdot 1 \cdot 36 = 144

= 1 3^{2} - 144

1 3^{2} = 169

= 169 - 144

Subtract the numbers:

169 - 144 = 25

= 25

Factor the number:

25 = 5^{2}

= 5^{2}

Apply radical rule:

5^{2} = 5

= 5

v_{1, 2} = \frac{- ( - 13 ) \pm 5}{2 \cdot 1}

Separate the solutions

v_{1} = \frac{- ( - 13 ) + 5}{2 \cdot 1}, v_{2} = \frac{- ( - 13 ) - 5}{2 \cdot 1}

v = \frac{- ( - 13 ) + 5}{2 \cdot 1} : 9

\frac{- ( - 13 ) + 5}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{13 + 5}{2 \cdot 1}

Add the numbers:

13 + 5 = 18

= \frac{18}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{18}{2}

Divide the numbers:

\frac{18}{2} = 9

= 9

v = \frac{- ( - 13 ) - 5}{2 \cdot 1} : 4

\frac{- ( - 13 ) - 5}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{13 - 5}{2 \cdot 1}

Subtract the numbers:

13 - 5 = 8

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{8}{2}

Divide the numbers:

\frac{8}{2} = 4

= 4

The solutions to the quadratic equation are:

v = 9, v = 4

v = 9, v = 4

Substitute back

v = u^{2},

solve for

u

Solve

u^{2} = 9 : u = 3, u = - 3

u^{2} = 9

For

x^{2} = f (a)

the solutions are

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

u = 9, u = - 9

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

- 9 = - 3

- 9

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

= - 3

u = 3, u = - 3

Solve

u^{2} = 4 : u = 2, u = - 2

u^{2} = 4

For

x^{2} = f (a)

the solutions are

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

u = 4, u = - 4

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

- 4 = - 2

- 4

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= - 2

u = 2, u = - 2

The solutions are

u = 3, u = - 3, u = 2, u = - 2

Substitute back

u = tan (x)

tan (x) = 3, tan (x) = - 3, tan (x) = 2, tan (x) = - 2

tan (x) = 3, tan (x) = - 3, tan (x) = 2, tan (x) = - 2

tan (x) = 3 : x = arctan (3) + πn

tan (x) = 3

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

tan (x) = - 3 : x = arctan (- 3) + πn

tan (x) = - 3

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

tan (x) = 2 : x = arctan (2) + πn

tan (x) = 2

Apply trig inverse properties

tan (x) = 2

General solutions for

tan (x) = 2

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

x = arctan (2) + πn

x = arctan (2) + πn

tan (x) = - 2 : x = arctan (- 2) + πn

tan (x) = - 2

Apply trig inverse properties

tan (x) = - 2

General solutions for

tan (x) = - 2

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

x = arctan (- 2) + πn

x = arctan (- 2) + πn

Combine all the solutions

x = arctan (3) + πn, x = arctan (- 3) + πn, x = arctan (2) + πn, x = arctan (- 2) + πn

Show solutions in decimal form

x = 1.24904 \dots + πn, x = - 1.24904 \dots + πn, x = 1.10714 \dots + πn, x = - 1.10714 \dots + πn

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