What is the general solution for tan^2(x)+7=-10sec(x),0<x<2pi ?

Q: What is the general solution for tan^2(x)+7=-10sec(x),0<x<2pi ?

The general solution for tan^2(x)+7=-10sec(x),0<x<2pi is x=1.67785…,x=-1.67785…+2pi

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Popular >

tan^2(x)+7=-10sec(x),0<x<2pi

Solution

tan^{2} (x) + 7 = - 10 sec (x), 0 < x < 2 π

Solution

x = 1.67785 \dots, x = - 1.67785 \dots + 2 π

Degrees

x = 96.13377 \dots^{\circ}, x = 263.86622 \dots^{\circ}

Solution steps

tan^{2} (x) + 7 = - 10 sec (x), 0 < x < 2 π

Subtract

- 10 sec (x)

from both sides

tan^{2} (x) + 7 + 10 sec (x) = 0

Rewrite using trig identities

7 + tan^{2} (x) + 10 sec (x)

Use the Pythagorean identity:

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

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

= 7 + sec^{2} (x) - 1 + 10 sec (x)

Simplify

7 + sec^{2} (x) - 1 + 10 sec (x) : sec^{2} (x) + 10 sec (x) + 6

7 + sec^{2} (x) - 1 + 10 sec (x)

Group like terms

= sec^{2} (x) + 10 sec (x) + 7 - 1

Add/Subtract the numbers:

7 - 1 = 6

= sec^{2} (x) + 10 sec (x) + 6

= sec^{2} (x) + 10 sec (x) + 6

6 + sec^{2} (x) + 10 sec (x) = 0

Solve by substitution

6 + sec^{2} (x) + 10 sec (x) = 0

Let:

sec (x) = u

6 + u^{2} + 10 u = 0

6 + u^{2} + 10 u = 0 : u = - 5 + 19, u = - 5 - 19

6 + u^{2} + 10 u = 0

Write in the standard form

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

u^{2} + 10 u + 6 = 0

Solve with the quadratic formula

u^{2} + 10 u + 6 = 0

Quadratic Equation Formula:

For

a = 1, b = 10, c = 6

u_{1, 2} = \frac{- 10 \pm 1 0 ^{2} - 4 \cdot 1 \cdot 6}{2 \cdot 1}

u_{1, 2} = \frac{- 10 \pm 1 0 ^{2} - 4 \cdot 1 \cdot 6}{2 \cdot 1}

1 0^{2} - 4 \cdot 1 \cdot 6 = 219

1 0^{2} - 4 \cdot 1 \cdot 6

Multiply the numbers:

4 \cdot 1 \cdot 6 = 24

= 1 0^{2} - 24

1 0^{2} = 100

= 100 - 24

Subtract the numbers:

100 - 24 = 76

= 76

Prime factorization of

76 : 2^{2} \cdot 19

76

76

divides by

276 = 38 \cdot 2

= 2 \cdot 38

38

divides by

238 = 19 \cdot 2

= 2 \cdot 2 \cdot 19

2, 19

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 19

= 2^{2} \cdot 19

= 2^{2} \cdot 19

Apply radical rule:

= 19 2^{2}

Apply radical rule:

2^{2} = 2

= 219

u_{1, 2} = \frac{- 10 \pm 2 19}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- 10 + 2 19}{2 \cdot 1}, u_{2} = \frac{- 10 - 2 19}{2 \cdot 1}

u = \frac{- 10 + 2 19}{2 \cdot 1} : - 5 + 19

\frac{- 10 + 2 19}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 10 + 2 19}{2}

Factor

- 10 + 219 : 2 (- 5 + 19)

- 10 + 219

Rewrite as

= - 2 \cdot 5 + 219

Factor out common term

2

= 2 (- 5 + 19)

= \frac{2 ( - 5 + 19 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= - 5 + 19

u = \frac{- 10 - 2 19}{2 \cdot 1} : - 5 - 19

\frac{- 10 - 2 19}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 10 - 2 19}{2}

Factor

- 10 - 219 : - 2 (5 + 19)

- 10 - 219

Rewrite as

= - 2 \cdot 5 - 219

Factor out common term

2

= - 2 (5 + 19)

= - \frac{2 ( 5 + 19 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= - (5 + 19)

Negate

- (5 + 19) = - 5 - 19

= - 5 - 19

The solutions to the quadratic equation are:

u = - 5 + 19, u = - 5 - 19

Substitute back

u = sec (x)

sec (x) = - 5 + 19, sec (x) = - 5 - 19

sec (x) = - 5 + 19, sec (x) = - 5 - 19

sec (x) = - 5 + 19, 0 < x < 2 π :

No Solution

sec (x) = - 5 + 19, 0 < x < 2 π

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

No Solution

sec (x) = - 5 - 19, 0 < x < 2 π : x = arcsec (- 5 - 19), x = - arcsec (- 5 - 19) + 2 π

sec (x) = - 5 - 19, 0 < x < 2 π

Apply trig inverse properties

sec (x) = - 5 - 19

General solutions for

sec (x) = - 5 - 19

sec (x) = - a \Rightarrow x = arcsec (- a) + 2 πn, x = - arcsec (- a) + 2 πn

x = arcsec (- 5 - 19) + 2 πn, x = - arcsec (- 5 - 19) + 2 πn

x = arcsec (- 5 - 19) + 2 πn, x = - arcsec (- 5 - 19) + 2 πn

Solutions for the range

0 < x < 2 π

x = arcsec (- 5 - 19), x = - arcsec (- 5 - 19) + 2 π

Combine all the solutions

x = arcsec (- 5 - 19), x = - arcsec (- 5 - 19) + 2 π

Show solutions in decimal form

x = 1.67785 \dots, x = - 1.67785 \dots + 2 π

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Frequently Asked Questions (FAQ)

What is the general solution for tan^2(x)+7=-10sec(x),0<x<2pi ?
The general solution for tan^2(x)+7=-10sec(x),0<x<2pi is x=1.67785…,x=-1.67785…+2pi