What is the general solution for 6tan^2(x)=5sec(x) ?

The general solution for 6tan^2(x)=5sec(x) is x=0.84106…+2pin,x=2pi-0.84106…+2pin

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6tan^2(x)=5sec(x)

Solution

6 tan^{2} (x) = 5 sec (x)

Solution

x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

Degrees

x = 48.18968 \dots^{\circ} + 36 0^{\circ} n, x = 311.81031 \dots^{\circ} + 36 0^{\circ} n

Solution steps

6 tan^{2} (x) = 5 sec (x)

Subtract

5 sec (x)

from both sides

6 tan^{2} (x) - 5 sec (x) = 0

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

= - 5 sec (x) + 6 (sec^{2} (x) - 1)

(- 1 + sec^{2} (x)) \cdot 6 - 5 sec (x) = 0

Solve by substitution

(- 1 + sec^{2} (x)) \cdot 6 - 5 sec (x) = 0

Let:

sec (x) = u

(- 1 + u^{2}) \cdot 6 - 5 u = 0

(- 1 + u^{2}) \cdot 6 - 5 u = 0 : u = \frac{3}{2}, u = - \frac{2}{3}

(- 1 + u^{2}) \cdot 6 - 5 u = 0

Expand

(- 1 + u^{2}) \cdot 6 - 5 u : - 6 + 6 u^{2} - 5 u

(- 1 + u^{2}) \cdot 6 - 5 u

= 6 (- 1 + u^{2}) - 5 u

Expand

6 (- 1 + u^{2}) : - 6 + 6 u^{2}

6 (- 1 + u^{2})

Apply the distributive law:

a (b + c) = ab + a c

a = 6, b = - 1, c = u^{2}

= 6 (- 1) + 6 u^{2}

Apply minus-plus rules

+ (- a) = - a

= - 6 \cdot 1 + 6 u^{2}

Multiply the numbers:

6 \cdot 1 = 6

= - 6 + 6 u^{2}

= - 6 + 6 u^{2} - 5 u

- 6 + 6 u^{2} - 5 u = 0

Write in the standard form

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

6 u^{2} - 5 u - 6 = 0

Solve with the quadratic formula

6 u^{2} - 5 u - 6 = 0

Quadratic Equation Formula:

For

a = 6, b = - 5, c = - 6

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 6 ( - 6 )}{2 \cdot 6}

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 6 ( - 6 )}{2 \cdot 6}

(- 5)^{2} - 4 \cdot 6 (- 6) = 13

(- 5)^{2} - 4 \cdot 6 (- 6)

Apply rule

- (- a) = a

= (- 5)^{2} + 4 \cdot 6 \cdot 6

Apply exponent rule:

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

n

is even

(- 5)^{2} = 5^{2}

= 5^{2} + 4 \cdot 6 \cdot 6

Multiply the numbers:

4 \cdot 6 \cdot 6 = 144

= 5^{2} + 144

5^{2} = 25

= 25 + 144

Add the numbers:

25 + 144 = 169

= 169

Factor the number:

169 = 1 3^{2}

= 1 3^{2}

Apply radical rule:

1 3^{2} = 13

= 13

u_{1, 2} = \frac{- ( - 5 ) \pm 13}{2 \cdot 6}

Separate the solutions

u_{1} = \frac{- ( - 5 ) + 13}{2 \cdot 6}, u_{2} = \frac{- ( - 5 ) - 13}{2 \cdot 6}

u = \frac{- ( - 5 ) + 13}{2 \cdot 6} : \frac{3}{2}

\frac{- ( - 5 ) + 13}{2 \cdot 6}

Apply rule

- (- a) = a

= \frac{5 + 13}{2 \cdot 6}

Add the numbers:

5 + 13 = 18

= \frac{18}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{18}{12}

Cancel the common factor:

6

= \frac{3}{2}

u = \frac{- ( - 5 ) - 13}{2 \cdot 6} : - \frac{2}{3}

\frac{- ( - 5 ) - 13}{2 \cdot 6}

Apply rule

- (- a) = a

= \frac{5 - 13}{2 \cdot 6}

Subtract the numbers:

5 - 13 = - 8

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

Multiply the numbers:

2 \cdot 6 = 12

= \frac{- 8}{12}

Apply the fraction rule:

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

= - \frac{8}{12}

Cancel the common factor:

4

= - \frac{2}{3}

The solutions to the quadratic equation are:

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

Substitute back

u = sec (x)

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

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

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

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

Apply trig inverse properties

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

General solutions for

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

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

x = arcsec (\frac{3}{2}) + 2 πn, x = 2 π - arcsec (\frac{3}{2}) + 2 πn

x = arcsec (\frac{3}{2}) + 2 πn, x = 2 π - arcsec (\frac{3}{2}) + 2 πn

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

No Solution

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

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

No Solution

Combine all the solutions

x = arcsec (\frac{3}{2}) + 2 πn, x = 2 π - arcsec (\frac{3}{2}) + 2 πn

Show solutions in decimal form

x = 0.84106 \dots + 2 πn, x = 2 π - 0.84106 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 6tan^2(x)=5sec(x) ?
The general solution for 6tan^2(x)=5sec(x) is x=0.84106…+2pin,x=2pi-0.84106…+2pin