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

The general solution for 2sec^2(x)=5tan(x)+5 is x=1.24904…+pin,x=-0.46364…+pin

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

Solution

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

Solution

x = 1.24904 \dots + πn, x = - 0.46364 \dots + πn

Degrees

x = 71.56505 \dots^{\circ} + 18 0^{\circ} n, x = - 26.56505 \dots^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

5 tan (x) + 5

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

Simplify

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

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

Expand

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

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

Apply the distributive law:

a (b + c) = ab + a c

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

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

Multiply the numbers:

2 \cdot 1 = 2

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

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

Simplify

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

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

Group like terms

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

Add/Subtract the numbers:

- 5 + 2 = - 3

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

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

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

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

Solve by substitution

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

Let:

tan (x) = u

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

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

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

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = 2, b = - 5, c = - 3

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

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

(- 5)^{2} - 4 \cdot 2 (- 3) = 7

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

Apply rule

- (- a) = a

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

Apply exponent rule:

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

n

is even

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

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

Multiply the numbers:

4 \cdot 2 \cdot 3 = 24

= 5^{2} + 24

5^{2} = 25

= 25 + 24

Add the numbers:

25 + 24 = 49

= 49

Factor the number:

49 = 7^{2}

= 7^{2}

Apply radical rule:

7^{2} = 7

= 7

u_{1, 2} = \frac{- ( - 5 ) \pm 7}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- ( - 5 ) + 7}{2 \cdot 2}, u_{2} = \frac{- ( - 5 ) - 7}{2 \cdot 2}

u = \frac{- ( - 5 ) + 7}{2 \cdot 2} : 3

\frac{- ( - 5 ) + 7}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{5 + 7}{2 \cdot 2}

Add the numbers:

5 + 7 = 12

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{12}{4}

Divide the numbers:

\frac{12}{4} = 3

= 3

u = \frac{- ( - 5 ) - 7}{2 \cdot 2} : - \frac{1}{2}

\frac{- ( - 5 ) - 7}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{5 - 7}{2 \cdot 2}

Subtract the numbers:

5 - 7 = - 2

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 2}{4}

Apply the fraction rule:

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

= - \frac{2}{4}

Cancel the common factor:

2

= - \frac{1}{2}

The solutions to the quadratic equation are:

u = 3, u = - \frac{1}{2}

Substitute back

u = tan (x)

tan (x) = 3, tan (x) = - \frac{1}{2}

tan (x) = 3, tan (x) = - \frac{1}{2}

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}{2} : x = arctan (- \frac{1}{2}) + πn

tan (x) = - \frac{1}{2}

Apply trig inverse properties

tan (x) = - \frac{1}{2}

General solutions for

tan (x) = - \frac{1}{2}

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

x = arctan (- \frac{1}{2}) + πn

x = arctan (- \frac{1}{2}) + πn

Combine all the solutions

x = arctan (3) + πn, x = arctan (- \frac{1}{2}) + πn

Show solutions in decimal form

x = 1.24904 \dots + πn, x = - 0.46364 \dots + πn

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Frequently Asked Questions (FAQ)

What is the general solution for 2sec^2(x)=5tan(x)+5 ?
The general solution for 2sec^2(x)=5tan(x)+5 is x=1.24904…+pin,x=-0.46364…+pin