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

The general solution for 2sec^2(x)+tan(x)=8 is x=0.98279…+pin,x=-1.10714…+pin

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Popular >

2sec^2(x)+tan(x)=8

Solution

2 sec^{2} (x) + tan (x) = 8

Solution

x = 0.98279 \dots + πn, x = - 1.10714 \dots + πn

Degrees

x = 56.30993 \dots^{\circ} + 18 0^{\circ} n, x = - 63.43494 \dots^{\circ} + 18 0^{\circ} n

Solution steps

2 sec^{2} (x) + tan (x) = 8

Subtract

8

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

Simplify

- 8 + tan (x) + 2 (tan^{2} (x) + 1) : 2 tan^{2} (x) + tan (x) - 6

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

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

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

Simplify

- 8 + tan (x) + 2 tan^{2} (x) + 2 : 2 tan^{2} (x) + tan (x) - 6

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

Group like terms

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

Add/Subtract the numbers:

- 8 + 2 = - 6

= 2 tan^{2} (x) + tan (x) - 6

= 2 tan^{2} (x) + tan (x) - 6

= 2 tan^{2} (x) + tan (x) - 6

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

Solve by substitution

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

Let:

tan (x) = u

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

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

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

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = 2, b = 1, c = - 6

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

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

1^{2} - 4 \cdot 2 (- 6) = 7

1^{2} - 4 \cdot 2 (- 6)

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 2 (- 6)

Apply rule

- (- a) = a

= 1 + 4 \cdot 2 \cdot 6

Multiply the numbers:

4 \cdot 2 \cdot 6 = 48

= 1 + 48

Add the numbers:

1 + 48 = 49

= 49

Factor the number:

49 = 7^{2}

= 7^{2}

Apply radical rule:

7^{2} = 7

= 7

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

Separate the solutions

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

u = \frac{- 1 + 7}{2 \cdot 2} : \frac{3}{2}

\frac{- 1 + 7}{2 \cdot 2}

Add/Subtract the numbers:

- 1 + 7 = 6

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{6}{4}

Cancel the common factor:

2

= \frac{3}{2}

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

\frac{- 1 - 7}{2 \cdot 2}

Subtract the numbers:

- 1 - 7 = - 8

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 8}{4}

Apply the fraction rule:

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

= - \frac{8}{4}

Divide the numbers:

\frac{8}{4} = 2

= - 2

The solutions to the quadratic equation are:

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

Substitute back

u = tan (x)

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

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

tan (x) = \frac{3}{2} : x = arctan (\frac{3}{2}) + πn

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

Apply trig inverse properties

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

General solutions for

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

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

x = arctan (\frac{3}{2}) + πn

x = arctan (\frac{3}{2}) + πn

tan (x) = - 2 : x = arctan (- 2) + πn

tan (x) = - 2

Apply trig inverse properties

tan (x) = - 2

General solutions for

tan (x) = - 2

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

x = arctan (- 2) + πn

x = arctan (- 2) + πn

Combine all the solutions

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

Show solutions in decimal form

x = 0.98279 \dots + πn, x = - 1.10714 \dots + πn

Popular Examples

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Frequently Asked Questions (FAQ)

What is the general solution for 2sec^2(x)+tan(x)=8 ?
The general solution for 2sec^2(x)+tan(x)=8 is x=0.98279…+pin,x=-1.10714…+pin