What is the general solution for 0=2tan(x)-sec^2(x) ?

The general solution for 0=2tan(x)-sec^2(x) is x= pi/4+pin

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

0=2tan(x)-sec^2(x)

Solution

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

Solution

x = \frac{π}{4} + πn

Degrees

x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

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

Switch sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

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

Distribute parentheses

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

Apply minus-plus rules

+ (- a) = - a

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

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

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

Solve by substitution

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

Let:

tan (x) = u

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

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

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

Write in the standard form

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

- u^{2} + 2 u - 1 = 0

Solve with the quadratic formula

- u^{2} + 2 u - 1 = 0

Quadratic Equation Formula:

For

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

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

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

2^{2} - 4 (- 1) (- 1) = 0

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

Apply rule

- (- a) = a

= 2^{2} - 4 \cdot 1 \cdot 1

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 2^{2} - 4

2^{2} = 4

= 4 - 4

Subtract the numbers:

4 - 4 = 0

= 0

u_{1, 2} = \frac{- 2 \pm 0}{2 ( - 1 )}

u = \frac{- 2}{2 ( - 1 )}

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

\frac{- 2}{2 ( - 1 )}

Remove parentheses:

(- a) = - a

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

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 2}{- 2}

Apply the fraction rule:

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

= \frac{2}{2}

Apply rule

\frac{a}{a} = 1

= 1

u = 1

The solution to the quadratic equation is:

u = 1

Substitute back

u = tan (x)

tan (x) = 1

tan (x) = 1

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

Combine all the solutions

x = \frac{π}{4} + πn

Popular Examples

csc^2(x)+6csc(x)-7=0

csc^{2} (x) + 6 csc (x) - 7 = 0

(sin(2x)-1)/3 =-1/6

\frac{sin ( 2 x ) - 1}{3} = - \frac{1}{6}

2sin^2(2x)+cos(2x)-1=0

2 sin^{2} (2 x) + cos (2 x) - 1 = 0

4cos(x)+2=3

4 cos (x) + 2 = 3

sin(4θ)=4sin(2θ)cos(2θ)

sin (4 θ) = 4 sin (2 θ) cos (2 θ)

Frequently Asked Questions (FAQ)

What is the general solution for 0=2tan(x)-sec^2(x) ?
The general solution for 0=2tan(x)-sec^2(x) is x= pi/4+pin