What is the general solution for 5sec^2(x)-14tan(x)=8 ?

The general solution for 5sec^2(x)-14tan(x)=8 is x=1.24904…+pin,x=-0.19739…+pin

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5sec^2(x)-14tan(x)=8

Solution

5 sec^{2} (x) - 14 tan (x) = 8

Solution

x = 1.24904 \dots + πn, x = - 0.19739 \dots + πn

Degrees

x = 71.56505 \dots^{\circ} + 18 0^{\circ} n, x = - 11.30993 \dots^{\circ} + 18 0^{\circ} n

Solution steps

5 sec^{2} (x) - 14 tan (x) = 8

Subtract

8

from both sides

5 sec^{2} (x) - 14 tan (x) - 8 = 0

Rewrite using trig identities

- 8 - 14 tan (x) + 5 sec^{2} (x)

Use the Pythagorean identity:

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

= - 8 - 14 tan (x) + 5 (tan^{2} (x) + 1)

Simplify

- 8 - 14 tan (x) + 5 (tan^{2} (x) + 1) : 5 tan^{2} (x) - 14 tan (x) - 3

- 8 - 14 tan (x) + 5 (tan^{2} (x) + 1)

Expand

5 (tan^{2} (x) + 1) : 5 tan^{2} (x) + 5

5 (tan^{2} (x) + 1)

Apply the distributive law:

a (b + c) = ab + a c

a = 5, b = tan^{2} (x), c = 1

= 5 tan^{2} (x) + 5 \cdot 1

Multiply the numbers:

5 \cdot 1 = 5

= 5 tan^{2} (x) + 5

= - 8 - 14 tan (x) + 5 tan^{2} (x) + 5

Simplify

- 8 - 14 tan (x) + 5 tan^{2} (x) + 5 : 5 tan^{2} (x) - 14 tan (x) - 3

- 8 - 14 tan (x) + 5 tan^{2} (x) + 5

Group like terms

= - 14 tan (x) + 5 tan^{2} (x) - 8 + 5

Add/Subtract the numbers:

- 8 + 5 = - 3

= 5 tan^{2} (x) - 14 tan (x) - 3

= 5 tan^{2} (x) - 14 tan (x) - 3

= 5 tan^{2} (x) - 14 tan (x) - 3

- 3 - 14 tan (x) + 5 tan^{2} (x) = 0

Solve by substitution

- 3 - 14 tan (x) + 5 tan^{2} (x) = 0

Let:

tan (x) = u

- 3 - 14 u + 5 u^{2} = 0

- 3 - 14 u + 5 u^{2} = 0 : u = 3, u = - \frac{1}{5}

- 3 - 14 u + 5 u^{2} = 0

Write in the standard form

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

5 u^{2} - 14 u - 3 = 0

Solve with the quadratic formula

5 u^{2} - 14 u - 3 = 0

Quadratic Equation Formula:

For

a = 5, b = - 14, c = - 3

u_{1, 2} = \frac{- ( - 14 ) \pm ( - 14 ) ^{2} - 4 \cdot 5 ( - 3 )}{2 \cdot 5}

u_{1, 2} = \frac{- ( - 14 ) \pm ( - 14 ) ^{2} - 4 \cdot 5 ( - 3 )}{2 \cdot 5}

(- 14)^{2} - 4 \cdot 5 (- 3) = 16

(- 14)^{2} - 4 \cdot 5 (- 3)

Apply rule

- (- a) = a

= (- 14)^{2} + 4 \cdot 5 \cdot 3

Apply exponent rule:

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

n

is even

(- 14)^{2} = 1 4^{2}

= 1 4^{2} + 4 \cdot 5 \cdot 3

Multiply the numbers:

4 \cdot 5 \cdot 3 = 60

= 1 4^{2} + 60

1 4^{2} = 196

= 196 + 60

Add the numbers:

196 + 60 = 256

= 256

Factor the number:

256 = 1 6^{2}

= 1 6^{2}

Apply radical rule:

1 6^{2} = 16

= 16

u_{1, 2} = \frac{- ( - 14 ) \pm 16}{2 \cdot 5}

Separate the solutions

u_{1} = \frac{- ( - 14 ) + 16}{2 \cdot 5}, u_{2} = \frac{- ( - 14 ) - 16}{2 \cdot 5}

u = \frac{- ( - 14 ) + 16}{2 \cdot 5} : 3

\frac{- ( - 14 ) + 16}{2 \cdot 5}

Apply rule

- (- a) = a

= \frac{14 + 16}{2 \cdot 5}

Add the numbers:

14 + 16 = 30

= \frac{30}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{30}{10}

Divide the numbers:

\frac{30}{10} = 3

= 3

u = \frac{- ( - 14 ) - 16}{2 \cdot 5} : - \frac{1}{5}

\frac{- ( - 14 ) - 16}{2 \cdot 5}

Apply rule

- (- a) = a

= \frac{14 - 16}{2 \cdot 5}

Subtract the numbers:

14 - 16 = - 2

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

Multiply the numbers:

2 \cdot 5 = 10

= \frac{- 2}{10}

Apply the fraction rule:

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

= - \frac{2}{10}

Cancel the common factor:

2

= - \frac{1}{5}

The solutions to the quadratic equation are:

u = 3, u = - \frac{1}{5}

Substitute back

u = tan (x)

tan (x) = 3, tan (x) = - \frac{1}{5}

tan (x) = 3, tan (x) = - \frac{1}{5}

tan (x) = 3 : x = arctan (3) + πn

tan (x) = 3

Apply trig inverse properties

tan (x) = 3

General solutions for

tan (x) = 3

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

x = arctan (3) + πn

x = arctan (3) + πn

tan (x) = - \frac{1}{5} : x = arctan (- \frac{1}{5}) + πn

tan (x) = - \frac{1}{5}

Apply trig inverse properties

tan (x) = - \frac{1}{5}

General solutions for

tan (x) = - \frac{1}{5}

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

x = arctan (- \frac{1}{5}) + πn

x = arctan (- \frac{1}{5}) + πn

Combine all the solutions

x = arctan (3) + πn, x = arctan (- \frac{1}{5}) + πn

Show solutions in decimal form

x = 1.24904 \dots + πn, x = - 0.19739 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 5sec^2(x)-14tan(x)=8 ?
The general solution for 5sec^2(x)-14tan(x)=8 is x=1.24904…+pin,x=-0.19739…+pin