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

The general solution for tan^2(x)-2=3tan(x) is x=1.29706…+pin,x=-0.51166…+pin

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

Solution

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

Solution

x = 1.29706 \dots + πn, x = - 0.51166 \dots + πn

Degrees

x = 74.31651 \dots^{\circ} + 18 0^{\circ} n, x = - 29.31651 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

tan (x) = u

u^{2} - 2 = 3 u

u^{2} - 2 = 3 u : u = \frac{3 + 17}{2}, u = \frac{3 - 17}{2}

u^{2} - 2 = 3 u

Move

3 u

to the left side

u^{2} - 2 = 3 u

Subtract

3 u

from both sides

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

Simplify

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

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

Write in the standard form

a x^{2} + b x + c = 0

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = 1, b = - 3, c = - 2

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

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

(- 3)^{2} - 4 \cdot 1 \cdot (- 2) = 17

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

Apply rule

- (- a) = a

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

Apply exponent rule:

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

n

is even

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

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

Multiply the numbers:

4 \cdot 1 \cdot 2 = 8

= 3^{2} + 8

3^{2} = 9

= 9 + 8

Add the numbers:

9 + 8 = 17

= 17

u_{1, 2} = \frac{- ( - 3 ) \pm 17}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- ( - 3 ) + 17}{2 \cdot 1}, u_{2} = \frac{- ( - 3 ) - 17}{2 \cdot 1}

u = \frac{- ( - 3 ) + 17}{2 \cdot 1} : \frac{3 + 17}{2}

\frac{- ( - 3 ) + 17}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{3 + 17}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{3 + 17}{2}

u = \frac{- ( - 3 ) - 17}{2 \cdot 1} : \frac{3 - 17}{2}

\frac{- ( - 3 ) - 17}{2 \cdot 1}

Apply rule

- (- a) = a

= \frac{3 - 17}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{3 - 17}{2}

The solutions to the quadratic equation are:

u = \frac{3 + 17}{2}, u = \frac{3 - 17}{2}

Substitute back

u = tan (x)

tan (x) = \frac{3 + 17}{2}, tan (x) = \frac{3 - 17}{2}

tan (x) = \frac{3 + 17}{2}, tan (x) = \frac{3 - 17}{2}

tan (x) = \frac{3 + 17}{2} : x = arctan (\frac{3 + 17}{2}) + πn

tan (x) = \frac{3 + 17}{2}

Apply trig inverse properties

tan (x) = \frac{3 + 17}{2}

General solutions for

tan (x) = \frac{3 + 17}{2}

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

x = arctan (\frac{3 + 17}{2}) + πn

x = arctan (\frac{3 + 17}{2}) + πn

tan (x) = \frac{3 - 17}{2} : x = arctan (\frac{3 - 17}{2}) + πn

tan (x) = \frac{3 - 17}{2}

Apply trig inverse properties

tan (x) = \frac{3 - 17}{2}

General solutions for

tan (x) = \frac{3 - 17}{2}

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

x = arctan (\frac{3 - 17}{2}) + πn

x = arctan (\frac{3 - 17}{2}) + πn

Combine all the solutions

x = arctan (\frac{3 + 17}{2}) + πn, x = arctan (\frac{3 - 17}{2}) + πn

Show solutions in decimal form

x = 1.29706 \dots + πn, x = - 0.51166 \dots + πn

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2 tan^{2} (θ) - 2 = 0

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2 csc (x) cos (x) = csc (x)

2cot^2(2x)=6

2 cot^{2} (2 x) = 6

Frequently Asked Questions (FAQ)

What is the general solution for tan^2(x)-2=3tan(x) ?
The general solution for tan^2(x)-2=3tan(x) is x=1.29706…+pin,x=-0.51166…+pin