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

The general solution for sec^2(x)+6tan(x)=8 is x= pi/4+pin,x=-1.42889…+pin

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

Solution

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

Solution

x = \frac{π}{4} + πn, x = - 1.42889 \dots + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n, x = - 81.86989 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

8

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

Simplify

- 8 + tan^{2} (x) + 1 + 6 tan (x) : tan^{2} (x) + 6 tan (x) - 7

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

Group like terms

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

Add/Subtract the numbers:

- 8 + 1 = - 7

= tan^{2} (x) + 6 tan (x) - 7

= tan^{2} (x) + 6 tan (x) - 7

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

Solve by substitution

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

Let:

tan (x) = u

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

- 7 + u^{2} + 6 u = 0 : u = 1, u = - 7

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

Write in the standard form

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

u^{2} + 6 u - 7 = 0

Solve with the quadratic formula

u^{2} + 6 u - 7 = 0

Quadratic Equation Formula:

For

a = 1, b = 6, c = - 7

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 \cdot 1 \cdot ( - 7 )}{2 \cdot 1}

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 \cdot 1 \cdot ( - 7 )}{2 \cdot 1}

6^{2} - 4 \cdot 1 \cdot (- 7) = 8

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

Apply rule

- (- a) = a

= 6^{2} + 4 \cdot 1 \cdot 7

Multiply the numbers:

4 \cdot 1 \cdot 7 = 28

= 6^{2} + 28

6^{2} = 36

= 36 + 28

Add the numbers:

36 + 28 = 64

= 64

Factor the number:

64 = 8^{2}

= 8^{2}

Apply radical rule:

n a^{n} = a

8^{2} = 8

= 8

u_{1, 2} = \frac{- 6 \pm 8}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- 6 + 8}{2 \cdot 1}, u_{2} = \frac{- 6 - 8}{2 \cdot 1}

u = \frac{- 6 + 8}{2 \cdot 1} : 1

\frac{- 6 + 8}{2 \cdot 1}

Add/Subtract the numbers:

- 6 + 8 = 2

= \frac{2}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

u = \frac{- 6 - 8}{2 \cdot 1} : - 7

\frac{- 6 - 8}{2 \cdot 1}

Subtract the numbers:

- 6 - 8 = - 14

= \frac{- 14}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 14}{2}

Apply the fraction rule:

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

= - \frac{14}{2}

Divide the numbers:

\frac{14}{2} = 7

= - 7

The solutions to the quadratic equation are:

u = 1, u = - 7

Substitute back

u = tan (x)

tan (x) = 1, tan (x) = - 7

tan (x) = 1, tan (x) = - 7

tan (x) = 1 : x = \frac{π}{4} + πn

tan (x) = 1

General solutions for

tan (x) = 1

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

tan (x) = - 7 : x = arctan (- 7) + πn

tan (x) = - 7

Apply trig inverse properties

tan (x) = - 7

General solutions for

tan (x) = - 7

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

x = arctan (- 7) + πn

x = arctan (- 7) + πn

Combine all the solutions

x = \frac{π}{4} + πn, x = arctan (- 7) + πn

Show solutions in decimal form

x = \frac{π}{4} + πn, x = - 1.42889 \dots + πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sec^2(x)+6tan(x)=8 ?
The general solution for sec^2(x)+6tan(x)=8 is x= pi/4+pin,x=-1.42889…+pin