What is the general solution for 8sec(x)-3tan^2(x)=7 ?

The general solution for 8sec(x)-3tan^2(x)=7 is x= pi/3+2pin,x=(5pi)/3+2pin

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8sec(x)-3tan^2(x)=7

Solution

8 sec (x) - 3 tan^{2} (x) = 7

Solution

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

8 sec (x) - 3 tan^{2} (x) = 7

Subtract

7

from both sides

8 sec (x) - 3 tan^{2} (x) - 7 = 0

Rewrite using trig identities

- 7 - 3 tan^{2} (x) + 8 sec (x)

Use the Pythagorean identity:

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

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

= - 7 - 3 (sec^{2} (x) - 1) + 8 sec (x)

Simplify

- 7 - 3 (sec^{2} (x) - 1) + 8 sec (x) : 8 sec (x) - 3 sec^{2} (x) - 4

- 7 - 3 (sec^{2} (x) - 1) + 8 sec (x)

Expand

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

- 3 (sec^{2} (x) - 1)

Apply the distributive law:

a (b - c) = ab - a c

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

= - 3 sec^{2} (x) - (- 3) \cdot 1

Apply minus-plus rules

- (- a) = a

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

Multiply the numbers:

3 \cdot 1 = 3

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

= - 7 - 3 sec^{2} (x) + 3 + 8 sec (x)

Simplify

- 7 - 3 sec^{2} (x) + 3 + 8 sec (x) : 8 sec (x) - 3 sec^{2} (x) - 4

- 7 - 3 sec^{2} (x) + 3 + 8 sec (x)

Group like terms

= - 3 sec^{2} (x) + 8 sec (x) - 7 + 3

Add/Subtract the numbers:

- 7 + 3 = - 4

= 8 sec (x) - 3 sec^{2} (x) - 4

= 8 sec (x) - 3 sec^{2} (x) - 4

= 8 sec (x) - 3 sec^{2} (x) - 4

- 4 - 3 sec^{2} (x) + 8 sec (x) = 0

Solve by substitution

- 4 - 3 sec^{2} (x) + 8 sec (x) = 0

Let:

sec (x) = u

- 4 - 3 u^{2} + 8 u = 0

- 4 - 3 u^{2} + 8 u = 0 : u = \frac{2}{3}, u = 2

- 4 - 3 u^{2} + 8 u = 0

Write in the standard form

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

- 3 u^{2} + 8 u - 4 = 0

Solve with the quadratic formula

- 3 u^{2} + 8 u - 4 = 0

Quadratic Equation Formula:

For

a = - 3, b = 8, c = - 4

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 ( - 3 ) ( - 4 )}{2 ( - 3 )}

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 ( - 3 ) ( - 4 )}{2 ( - 3 )}

8^{2} - 4 (- 3) (- 4) = 4

8^{2} - 4 (- 3) (- 4)

Apply rule

- (- a) = a

= 8^{2} - 4 \cdot 3 \cdot 4

Multiply the numbers:

4 \cdot 3 \cdot 4 = 48

= 8^{2} - 48

8^{2} = 64

= 64 - 48

Subtract the numbers:

64 - 48 = 16

= 16

Factor the number:

16 = 4^{2}

= 4^{2}

Apply radical rule:

4^{2} = 4

= 4

u_{1, 2} = \frac{- 8 \pm 4}{2 ( - 3 )}

Separate the solutions

u_{1} = \frac{- 8 + 4}{2 ( - 3 )}, u_{2} = \frac{- 8 - 4}{2 ( - 3 )}

u = \frac{- 8 + 4}{2 ( - 3 )} : \frac{2}{3}

\frac{- 8 + 4}{2 ( - 3 )}

Remove parentheses:

(- a) = - a

= \frac{- 8 + 4}{- 2 \cdot 3}

Add/Subtract the numbers:

- 8 + 4 = - 4

= \frac{- 4}{- 2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 4}{- 6}

Apply the fraction rule:

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

= \frac{4}{6}

Cancel the common factor:

2

= \frac{2}{3}

u = \frac{- 8 - 4}{2 ( - 3 )} : 2

\frac{- 8 - 4}{2 ( - 3 )}

Remove parentheses:

(- a) = - a

= \frac{- 8 - 4}{- 2 \cdot 3}

Subtract the numbers:

- 8 - 4 = - 12

= \frac{- 12}{- 2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 12}{- 6}

Apply the fraction rule:

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

= \frac{12}{6}

Divide the numbers:

\frac{12}{6} = 2

= 2

The solutions to the quadratic equation are:

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

Substitute back

u = sec (x)

sec (x) = \frac{2}{3}, sec (x) = 2

sec (x) = \frac{2}{3}, sec (x) = 2

sec (x) = \frac{2}{3} :

No Solution

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

sec (x) \leq - 1 or sec (x) \geq 1

No Solution

sec (x) = 2 : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

sec (x) = 2

General solutions for

sec (x) = 2

sec (x)

periodicity table with

2 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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combine all the solutions

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 8sec(x)-3tan^2(x)=7 ?
The general solution for 8sec(x)-3tan^2(x)=7 is x= pi/3+2pin,x=(5pi)/3+2pin