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Popular >

2sec(x)+tan^2(x)-9=1

Solution

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

Solution

x = 1.15291 \dots + 2 πn, x = 2 π - 1.15291 \dots + 2 πn, x = 1.79672 \dots + 2 πn, x = - 1.79672 \dots + 2 πn

Degrees

x = 66.05701 \dots^{\circ} + 36 0^{\circ} n, x = 293.94298 \dots^{\circ} + 36 0^{\circ} n, x = 102.94462 \dots^{\circ} + 36 0^{\circ} n, x = - 102.94462 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

1

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

- 10 + sec^{2} (x) - 1 + 2 sec (x) : sec^{2} (x) + 2 sec (x) - 11

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

Group like terms

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

Subtract the numbers:

- 10 - 1 = - 11

= sec^{2} (x) + 2 sec (x) - 11

= sec^{2} (x) + 2 sec (x) - 11

- 11 + sec^{2} (x) + 2 sec (x) = 0

Solve by substitution

- 11 + sec^{2} (x) + 2 sec (x) = 0

Let:

sec (x) = u

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

- 11 + u^{2} + 2 u = 0 : u = - 1 + 23, u = - 1 - 23

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

Write in the standard form

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

u^{2} + 2 u - 11 = 0

Solve with the quadratic formula

u^{2} + 2 u - 11 = 0

Quadratic Equation Formula:

For

a = 1, b = 2, c = - 11

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 \cdot 1 \cdot ( - 11 )}{2 \cdot 1}

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 \cdot 1 \cdot ( - 11 )}{2 \cdot 1}

2^{2} - 4 \cdot 1 \cdot (- 11) = 43

2^{2} - 4 \cdot 1 \cdot (- 11)

Apply rule

- (- a) = a

= 2^{2} + 4 \cdot 1 \cdot 11

Multiply the numbers:

4 \cdot 1 \cdot 11 = 44

= 2^{2} + 44

2^{2} = 4

= 4 + 44

Add the numbers:

4 + 44 = 48

= 48

Prime factorization of

48 : 2^{4} \cdot 3

48

48

divides by

248 = 24 \cdot 2

= 2 \cdot 24

24

divides by

224 = 12 \cdot 2

= 2 \cdot 2 \cdot 12

12

divides by

212 = 6 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3

= 2^{4} \cdot 3

= 2^{4} \cdot 3

Apply radical rule:

= 3 2^{4}

Apply radical rule:

2^{4} = 2^{\frac{4}{2}} = 2^{2}

= 2^{2} 3

Refine

= 43

u_{1, 2} = \frac{- 2 \pm 4 3}{2 \cdot 1}

Separate the solutions

u_{1} = \frac{- 2 + 4 3}{2 \cdot 1}, u_{2} = \frac{- 2 - 4 3}{2 \cdot 1}

u = \frac{- 2 + 4 3}{2 \cdot 1} : - 1 + 23

\frac{- 2 + 4 3}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

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

Factor

- 2 + 43 : 2 (- 1 + 23)

- 2 + 43

Rewrite as

= - 2 \cdot 1 + 2 \cdot 23

Factor out common term

2

= 2 (- 1 + 23)

= \frac{2 ( - 1 + 2 3 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= - 1 + 23

u = \frac{- 2 - 4 3}{2 \cdot 1} : - 1 - 23

\frac{- 2 - 4 3}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

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

Factor

- 2 - 43 : - 2 (1 + 23)

- 2 - 43

Rewrite as

= - 2 \cdot 1 - 2 \cdot 23

Factor out common term

2

= - 2 (1 + 23)

= - \frac{2 ( 1 + 2 3 )}{2}

Divide the numbers:

\frac{2}{2} = 1

= - (1 + 23)

Negate

- (1 + 23) = - 1 - 23

= - 1 - 23

The solutions to the quadratic equation are:

u = - 1 + 23, u = - 1 - 23

Substitute back

u = sec (x)

sec (x) = - 1 + 23, sec (x) = - 1 - 23

sec (x) = - 1 + 23, sec (x) = - 1 - 23

sec (x) = - 1 + 23 : x = arcsec (- 1 + 23) + 2 πn, x = 2 π - arcsec (- 1 + 23) + 2 πn

sec (x) = - 1 + 23

Apply trig inverse properties

sec (x) = - 1 + 23

General solutions for

sec (x) = - 1 + 23

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

x = arcsec (- 1 + 23) + 2 πn, x = 2 π - arcsec (- 1 + 23) + 2 πn

x = arcsec (- 1 + 23) + 2 πn, x = 2 π - arcsec (- 1 + 23) + 2 πn

sec (x) = - 1 - 23 : x = arcsec (- 1 - 23) + 2 πn, x = - arcsec (- 1 - 23) + 2 πn

sec (x) = - 1 - 23

Apply trig inverse properties

sec (x) = - 1 - 23

General solutions for

sec (x) = - 1 - 23

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

x = arcsec (- 1 - 23) + 2 πn, x = - arcsec (- 1 - 23) + 2 πn

x = arcsec (- 1 - 23) + 2 πn, x = - arcsec (- 1 - 23) + 2 πn

Combine all the solutions

x = arcsec (- 1 + 23) + 2 πn, x = 2 π - arcsec (- 1 + 23) + 2 πn, x = arcsec (- 1 - 23) + 2 πn, x = - arcsec (- 1 - 23) + 2 πn

Show solutions in decimal form

x = 1.15291 \dots + 2 πn, x = 2 π - 1.15291 \dots + 2 πn, x = 1.79672 \dots + 2 πn, x = - 1.79672 \dots + 2 πn

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