What is the general solution for 3sec^2(x)+tan(x)-6=0 ?

The general solution for 3sec^2(x)+tan(x)-6=0 is x=0.70282…+pin,x=-0.86797…+pin

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3sec^2(x)+tan(x)-6=0

Solution

3 sec^{2} (x) + tan (x) - 6 = 0

Solution

x = 0.70282 \dots + πn, x = - 0.86797 \dots + πn

Degrees

x = 40.26883 \dots^{\circ} + 18 0^{\circ} n, x = - 49.73116 \dots^{\circ} + 18 0^{\circ} n

Solution steps

3 sec^{2} (x) + tan (x) - 6 = 0

Rewrite using trig identities

- 6 + tan (x) + 3 sec^{2} (x)

Use the Pythagorean identity:

sec^{2} (x) = tan^{2} (x) + 1

= - 6 + tan (x) + 3 (tan^{2} (x) + 1)

Simplify

- 6 + tan (x) + 3 (tan^{2} (x) + 1) : 3 tan^{2} (x) + tan (x) - 3

- 6 + tan (x) + 3 (tan^{2} (x) + 1)

Expand

3 (tan^{2} (x) + 1) : 3 tan^{2} (x) + 3

3 (tan^{2} (x) + 1)

Apply the distributive law:

a (b + c) = ab + a c

a = 3, b = tan^{2} (x), c = 1

= 3 tan^{2} (x) + 3 \cdot 1

Multiply the numbers:

3 \cdot 1 = 3

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

= - 6 + tan (x) + 3 tan^{2} (x) + 3

Simplify

- 6 + tan (x) + 3 tan^{2} (x) + 3 : 3 tan^{2} (x) + tan (x) - 3

- 6 + tan (x) + 3 tan^{2} (x) + 3

Group like terms

= tan (x) + 3 tan^{2} (x) - 6 + 3

Add/Subtract the numbers:

- 6 + 3 = - 3

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

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

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

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

Solve by substitution

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

Let:

tan (x) = u

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

- 3 + u + 3 u^{2} = 0 : u = \frac{- 1 + 37}{6}, u = \frac{- 1 - 37}{6}

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

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

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

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

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

1^{2} - 4 \cdot 3 (- 3) = 37

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

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 3 (- 3)

Apply rule

- (- a) = a

= 1 + 4 \cdot 3 \cdot 3

Multiply the numbers:

4 \cdot 3 \cdot 3 = 36

= 1 + 36

Add the numbers:

1 + 36 = 37

= 37

u_{1, 2} = \frac{- 1 \pm 37}{2 \cdot 3}

Separate the solutions

u_{1} = \frac{- 1 + 37}{2 \cdot 3}, u_{2} = \frac{- 1 - 37}{2 \cdot 3}

u = \frac{- 1 + 37}{2 \cdot 3} : \frac{- 1 + 37}{6}

\frac{- 1 + 37}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 1 + 37}{6}

u = \frac{- 1 - 37}{2 \cdot 3} : \frac{- 1 - 37}{6}

\frac{- 1 - 37}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 1 - 37}{6}

The solutions to the quadratic equation are:

u = \frac{- 1 + 37}{6}, u = \frac{- 1 - 37}{6}

Substitute back

u = tan (x)

tan (x) = \frac{- 1 + 37}{6}, tan (x) = \frac{- 1 - 37}{6}

tan (x) = \frac{- 1 + 37}{6}, tan (x) = \frac{- 1 - 37}{6}

tan (x) = \frac{- 1 + 37}{6} : x = arctan (\frac{- 1 + 37}{6}) + πn

tan (x) = \frac{- 1 + 37}{6}

Apply trig inverse properties

tan (x) = \frac{- 1 + 37}{6}

General solutions for

tan (x) = \frac{- 1 + 37}{6}

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

x = arctan (\frac{- 1 + 37}{6}) + πn

x = arctan (\frac{- 1 + 37}{6}) + πn

tan (x) = \frac{- 1 - 37}{6} : x = arctan (\frac{- 1 - 37}{6}) + πn

tan (x) = \frac{- 1 - 37}{6}

Apply trig inverse properties

tan (x) = \frac{- 1 - 37}{6}

General solutions for

tan (x) = \frac{- 1 - 37}{6}

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

x = arctan (\frac{- 1 - 37}{6}) + πn

x = arctan (\frac{- 1 - 37}{6}) + πn

Combine all the solutions

x = arctan (\frac{- 1 + 37}{6}) + πn, x = arctan (\frac{- 1 - 37}{6}) + πn

Show solutions in decimal form

x = 0.70282 \dots + πn, x = - 0.86797 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 3sec^2(x)+tan(x)-6=0 ?
The general solution for 3sec^2(x)+tan(x)-6=0 is x=0.70282…+pin,x=-0.86797…+pin