What is the general solution for 2tan(x)+sec^2(x)=4 ?

The general solution for 2tan(x)+sec^2(x)=4 is x= pi/4+pin,x=-1.24904…+pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

2tan(x)+sec^2(x)=4

Solution

2 tan (x) + sec^{2} (x) = 4

Solution

x = \frac{π}{4} + πn, x = - 1.24904 \dots + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n, x = - 71.56505 \dots^{\circ} + 18 0^{\circ} n

Solution steps

2 tan (x) + sec^{2} (x) = 4

Subtract

4

from both sides

2 tan (x) + sec^{2} (x) - 4 = 0

Rewrite using trig identities

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

Use the Pythagorean identity:

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

= - 4 + tan^{2} (x) + 1 + 2 tan (x)

Simplify

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

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

Group like terms

= tan^{2} (x) + 2 tan (x) - 4 + 1

Add/Subtract the numbers:

- 4 + 1 = - 3

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

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

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

Solve by substitution

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

Let:

tan (x) = u

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

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

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

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = 1, b = 2, c = - 3

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

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

2^{2} - 4 \cdot 1 \cdot (- 3) = 4

2^{2} - 4 \cdot 1 \cdot (- 3)

Apply rule

- (- a) = a

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

Multiply the numbers:

4 \cdot 1 \cdot 3 = 12

= 2^{2} + 12

2^{2} = 4

= 4 + 12

Add the numbers:

4 + 12 = 16

= 16

Factor the number:

16 = 4^{2}

= 4^{2}

Apply radical rule:

4^{2} = 4

= 4

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

Separate the solutions

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

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

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

Add/Subtract the numbers:

- 2 + 4 = 2

= \frac{2}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

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

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

Subtract the numbers:

- 2 - 4 = - 6

= \frac{- 6}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 6}{2}

Apply the fraction rule:

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

= - \frac{6}{2}

Divide the numbers:

\frac{6}{2} = 3

= - 3

The solutions to the quadratic equation are:

u = 1, u = - 3

Substitute back

u = tan (x)

tan (x) = 1, tan (x) = - 3

tan (x) = 1, tan (x) = - 3

tan (x) = 1 : x = \frac{π}{4} + πn

tan (x) = 1

General solutions for

tan (x) = 1

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

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

Combine all the solutions

x = \frac{π}{4} + πn, x = arctan (- 3) + πn

Show solutions in decimal form

x = \frac{π}{4} + πn, x = - 1.24904 \dots + πn

Popular Examples

cos(a)+sin(a)=1

cos (a) + sin (a) = 1

cos(x)= 1/7

cos (x) = \frac{1}{7}

sin(5x+43)=cos(-x+31)

sin (5 x + 43) = cos (- x + 3 1^{\circ})

sin(x)+1=2cos^2(x)

sin (x) + 1 = 2 cos^{2} (x)

solvefor x,arctan(x^2+9y^2-2x-36y+37)=0 solve for

x, arctan (x^{2} + 9 y^{2} - 2 x - 36 y + 37) = 0

Frequently Asked Questions (FAQ)

What is the general solution for 2tan(x)+sec^2(x)=4 ?
The general solution for 2tan(x)+sec^2(x)=4 is x= pi/4+pin,x=-1.24904…+pin