What is the general solution for 6sec^2(x)-8=tan(x) ?

The general solution for 6sec^2(x)-8=tan(x) is x=0.58800…+pin,x=-0.46364…+pin

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Popular Trigonometry >

6sec^2(x)-8=tan(x)

Solution

6 sec^{2} (x) - 8 = tan (x)

Solution

x = 0.58800 \dots + πn, x = - 0.46364 \dots + πn

Degrees

x = 33.69006 \dots^{\circ} + 18 0^{\circ} n, x = - 26.56505 \dots^{\circ} + 18 0^{\circ} n

Solution steps

6 sec^{2} (x) - 8 = tan (x)

Subtract

tan (x)

from both sides

6 sec^{2} (x) - 8 - tan (x) = 0

Rewrite using trig identities

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

Use the Pythagorean identity:

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

= - 8 - tan (x) + 6 (tan^{2} (x) + 1)

Simplify

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

- 8 - 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

= - 8 - tan (x) + 6 tan^{2} (x) + 6

Simplify

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

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

Group like terms

= - tan (x) + 6 tan^{2} (x) - 8 + 6

Add/Subtract the numbers:

- 8 + 6 = - 2

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

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

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

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

Solve by substitution

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

Let:

tan (x) = u

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

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

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

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = 6, b = - 1, c = - 2

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

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

(- 1)^{2} - 4 \cdot 6 (- 2) = 7

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

Apply rule

- (- a) = a

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

(- 1)^{2} = 1

(- 1)^{2}

Apply exponent rule:

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

n

is even

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

= 1^{2}

Apply rule

1^{a} = 1

= 1

4 \cdot 6 \cdot 2 = 48

4 \cdot 6 \cdot 2

Multiply the numbers:

4 \cdot 6 \cdot 2 = 48

= 48

= 1 + 48

Add the numbers:

1 + 48 = 49

= 49

Factor the number:

49 = 7^{2}

= 7^{2}

Apply radical rule:

7^{2} = 7

= 7

u_{1, 2} = \frac{- ( - 1 ) \pm 7}{2 \cdot 6}

Separate the solutions

u_{1} = \frac{- ( - 1 ) + 7}{2 \cdot 6}, u_{2} = \frac{- ( - 1 ) - 7}{2 \cdot 6}

u = \frac{- ( - 1 ) + 7}{2 \cdot 6} : \frac{2}{3}

\frac{- ( - 1 ) + 7}{2 \cdot 6}

Apply rule

- (- a) = a

= \frac{1 + 7}{2 \cdot 6}

Add the numbers:

1 + 7 = 8

= \frac{8}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{8}{12}

Cancel the common factor:

4

= \frac{2}{3}

u = \frac{- ( - 1 ) - 7}{2 \cdot 6} : - \frac{1}{2}

\frac{- ( - 1 ) - 7}{2 \cdot 6}

Apply rule

- (- a) = a

= \frac{1 - 7}{2 \cdot 6}

Subtract the numbers:

1 - 7 = - 6

= \frac{- 6}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{- 6}{12}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{6}{12}

Cancel the common factor:

6

= - \frac{1}{2}

The solutions to the quadratic equation are:

u = \frac{2}{3}, u = - \frac{1}{2}

Substitute back

u = tan (x)

tan (x) = \frac{2}{3}, tan (x) = - \frac{1}{2}

tan (x) = \frac{2}{3}, tan (x) = - \frac{1}{2}

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

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

Apply trig inverse properties

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

General solutions for

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

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

x = arctan (\frac{2}{3}) + πn

x = arctan (\frac{2}{3}) + πn

tan (x) = - \frac{1}{2} : x = arctan (- \frac{1}{2}) + πn

tan (x) = - \frac{1}{2}

Apply trig inverse properties

tan (x) = - \frac{1}{2}

General solutions for

tan (x) = - \frac{1}{2}

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

x = arctan (- \frac{1}{2}) + πn

x = arctan (- \frac{1}{2}) + πn

Combine all the solutions

x = arctan (\frac{2}{3}) + πn, x = arctan (- \frac{1}{2}) + πn

Show solutions in decimal form

x = 0.58800 \dots + πn, x = - 0.46364 \dots + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 6sec^2(x)-8=tan(x) ?
The general solution for 6sec^2(x)-8=tan(x) is x=0.58800…+pin,x=-0.46364…+pin