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Popular >

tan^4(x)-3tan^2(x)+1=0

Solution

tan^{4} (x) - 3 tan^{2} (x) + 1 = 0

Solution

x = 1.01722 \dots + πn, x = - 1.01722 \dots + πn, x = 0.55357 \dots + πn, x = - 0.55357 \dots + πn

Degrees

x = 58.28252 \dots^{\circ} + 18 0^{\circ} n, x = - 58.28252 \dots^{\circ} + 18 0^{\circ} n, x = 31.71747 \dots^{\circ} + 18 0^{\circ} n, x = - 31.71747 \dots^{\circ} + 18 0^{\circ} n

Solution steps

tan^{4} (x) - 3 tan^{2} (x) + 1 = 0

Solve by substitution

tan^{4} (x) - 3 tan^{2} (x) + 1 = 0

Let:

tan (x) = u

u^{4} - 3 u^{2} + 1 = 0

u^{4} - 3 u^{2} + 1 = 0 : u = \frac{3 + 5}{2}, u = - \frac{3 + 5}{2}, u = \frac{3 - 5}{2}, u = - \frac{3 - 5}{2}

u^{4} - 3 u^{2} + 1 = 0

Rewrite the equation with

v = u^{2}

and

v^{2} = u^{4}

v^{2} - 3 v + 1 = 0

Solve

v^{2} - 3 v + 1 = 0 : v = \frac{3 + 5}{2}, v = \frac{3 - 5}{2}

v^{2} - 3 v + 1 = 0

Solve with the quadratic formula

v^{2} - 3 v + 1 = 0

Quadratic Equation Formula:

For

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

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

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

(- 3)^{2} - 4 \cdot 1 \cdot 1 = 5

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

Apply exponent rule:

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

n

is even

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

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

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 3^{2} - 4

3^{2} = 9

= 9 - 4

Subtract the numbers:

9 - 4 = 5

= 5

v_{1, 2} = \frac{- ( - 3 ) \pm 5}{2 \cdot 1}

Separate the solutions

v_{1} = \frac{- ( - 3 ) + 5}{2 \cdot 1}, v_{2} = \frac{- ( - 3 ) - 5}{2 \cdot 1}

v = \frac{- ( - 3 ) + 5}{2 \cdot 1} : \frac{3 + 5}{2}

\frac{- ( - 3 ) + 5}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{3 + 5}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{3 + 5}{2}

v = \frac{- ( - 3 ) - 5}{2 \cdot 1} : \frac{3 - 5}{2}

\frac{- ( - 3 ) - 5}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{3 - 5}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{3 - 5}{2}

The solutions to the quadratic equation are:

v = \frac{3 + 5}{2}, v = \frac{3 - 5}{2}

v = \frac{3 + 5}{2}, v = \frac{3 - 5}{2}

Substitute back

v = u^{2},

solve for

u

Solve

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

u^{2} = \frac{3 + 5}{2}

For

x^{2} = f (a)

the solutions are

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

u = \frac{3 + 5}{2}, u = - \frac{3 + 5}{2}

Solve

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

u^{2} = \frac{3 - 5}{2}

For

x^{2} = f (a)

the solutions are

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

u = \frac{3 - 5}{2}, u = - \frac{3 - 5}{2}

The solutions are

u = \frac{3 + 5}{2}, u = - \frac{3 + 5}{2}, u = \frac{3 - 5}{2}, u = - \frac{3 - 5}{2}

Substitute back

u = tan (x)

tan (x) = \frac{3 + 5}{2}, tan (x) = - \frac{3 + 5}{2}, tan (x) = \frac{3 - 5}{2}, tan (x) = - \frac{3 - 5}{2}

tan (x) = \frac{3 + 5}{2}, tan (x) = - \frac{3 + 5}{2}, tan (x) = \frac{3 - 5}{2}, tan (x) = - \frac{3 - 5}{2}

tan (x) = \frac{3 + 5}{2} : x = arctan \frac{3 + 5}{2} + πn

tan (x) = \frac{3 + 5}{2}

Apply trig inverse properties

tan (x) = \frac{3 + 5}{2}

General solutions for

tan (x) = \frac{3 + 5}{2}

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

x = arctan \frac{3 + 5}{2} + πn

x = arctan \frac{3 + 5}{2} + πn

tan (x) = - \frac{3 + 5}{2} : x = arctan - \frac{3 + 5}{2} + πn

tan (x) = - \frac{3 + 5}{2}

Apply trig inverse properties

tan (x) = - \frac{3 + 5}{2}

General solutions for

tan (x) = - \frac{3 + 5}{2}

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

x = arctan - \frac{3 + 5}{2} + πn

x = arctan - \frac{3 + 5}{2} + πn

tan (x) = \frac{3 - 5}{2} : x = arctan \frac{3 - 5}{2} + πn

tan (x) = \frac{3 - 5}{2}

Apply trig inverse properties

tan (x) = \frac{3 - 5}{2}

General solutions for

tan (x) = \frac{3 - 5}{2}

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

x = arctan \frac{3 - 5}{2} + πn

x = arctan \frac{3 - 5}{2} + πn

tan (x) = - \frac{3 - 5}{2} : x = arctan - \frac{3 - 5}{2} + πn

tan (x) = - \frac{3 - 5}{2}

Apply trig inverse properties

tan (x) = - \frac{3 - 5}{2}

General solutions for

tan (x) = - \frac{3 - 5}{2}

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

x = arctan - \frac{3 - 5}{2} + πn

x = arctan - \frac{3 - 5}{2} + πn

Combine all the solutions

x = arctan \frac{3 + 5}{2} + πn, x = arctan - \frac{3 + 5}{2} + πn, x = arctan \frac{3 - 5}{2} + πn, x = arctan - \frac{3 - 5}{2} + πn

Show solutions in decimal form

x = 1.01722 \dots + πn, x = - 1.01722 \dots + πn, x = 0.55357 \dots + πn, x = - 0.55357 \dots + πn

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