What is the general solution for 6sec^2(x)+tan(x)-9=0 ?

The general solution for 6sec^2(x)+tan(x)-9=0 is x=0.56123…+pin,x=-0.67188…+pin

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6sec^2(x)+tan(x)-9=0

Solution

6 sec^{2} (x) + tan (x) - 9 = 0

Solution

x = 0.56123 \dots + πn, x = - 0.67188 \dots + πn

Degrees

x = 32.15621 \dots^{\circ} + 18 0^{\circ} n, x = - 38.49641 \dots^{\circ} + 18 0^{\circ} n

Solution steps

6 sec^{2} (x) + tan (x) - 9 = 0

Rewrite using trig identities

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

Use the Pythagorean identity:

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

= - 9 + tan (x) + 6 (tan^{2} (x) + 1)

Simplify

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

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

Expand

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

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

Apply the distributive law:

a (b + c) = ab + a c

a = 6, b = tan^{2} (x), c = 1

= 6 tan^{2} (x) + 6 \cdot 1

Multiply the numbers:

6 \cdot 1 = 6

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

= - 9 + tan (x) + 6 tan^{2} (x) + 6

Simplify

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

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

Group like terms

= tan (x) + 6 tan^{2} (x) - 9 + 6

Add/Subtract the numbers:

- 9 + 6 = - 3

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

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

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

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

Solve by substitution

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

Let:

tan (x) = u

- 3 + u + 6 u^{2} = 0

- 3 + u + 6 u^{2} = 0 : u = \frac{- 1 + 73}{12}, u = \frac{- 1 - 73}{12}

- 3 + u + 6 u^{2} = 0

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = 6, b = 1, c = - 3

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 6 ( - 3 )}{2 \cdot 6}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 6 ( - 3 )}{2 \cdot 6}

1^{2} - 4 \cdot 6 (- 3) = 73

1^{2} - 4 \cdot 6 (- 3)

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 6 (- 3)

Apply rule

- (- a) = a

= 1 + 4 \cdot 6 \cdot 3

Multiply the numbers:

4 \cdot 6 \cdot 3 = 72

= 1 + 72

Add the numbers:

1 + 72 = 73

= 73

u_{1, 2} = \frac{- 1 \pm 73}{2 \cdot 6}

Separate the solutions

u_{1} = \frac{- 1 + 73}{2 \cdot 6}, u_{2} = \frac{- 1 - 73}{2 \cdot 6}

u = \frac{- 1 + 73}{2 \cdot 6} : \frac{- 1 + 73}{12}

\frac{- 1 + 73}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{- 1 + 73}{12}

u = \frac{- 1 - 73}{2 \cdot 6} : \frac{- 1 - 73}{12}

\frac{- 1 - 73}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{- 1 - 73}{12}

The solutions to the quadratic equation are:

u = \frac{- 1 + 73}{12}, u = \frac{- 1 - 73}{12}

Substitute back

u = tan (x)

tan (x) = \frac{- 1 + 73}{12}, tan (x) = \frac{- 1 - 73}{12}

tan (x) = \frac{- 1 + 73}{12}, tan (x) = \frac{- 1 - 73}{12}

tan (x) = \frac{- 1 + 73}{12} : x = arctan (\frac{- 1 + 73}{12}) + πn

tan (x) = \frac{- 1 + 73}{12}

Apply trig inverse properties

tan (x) = \frac{- 1 + 73}{12}

General solutions for

tan (x) = \frac{- 1 + 73}{12}

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

x = arctan (\frac{- 1 + 73}{12}) + πn

x = arctan (\frac{- 1 + 73}{12}) + πn

tan (x) = \frac{- 1 - 73}{12} : x = arctan (\frac{- 1 - 73}{12}) + πn

tan (x) = \frac{- 1 - 73}{12}

Apply trig inverse properties

tan (x) = \frac{- 1 - 73}{12}

General solutions for

tan (x) = \frac{- 1 - 73}{12}

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

x = arctan (\frac{- 1 - 73}{12}) + πn

x = arctan (\frac{- 1 - 73}{12}) + πn

Combine all the solutions

x = arctan (\frac{- 1 + 73}{12}) + πn, x = arctan (\frac{- 1 - 73}{12}) + πn

Show solutions in decimal form

x = 0.56123 \dots + πn, x = - 0.67188 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 6sec^2(x)+tan(x)-9=0 ?
The general solution for 6sec^2(x)+tan(x)-9=0 is x=0.56123…+pin,x=-0.67188…+pin